Introduction to Micrometeorology, Second Edition (International Geophysics)

Introduction to Micrometeorology Second Edition

This is Volume 79 in the INTERNATIONAL GEOPHYSICS SERIES A series of monographs and textbooks Edited by RENATA DMOWSKA, JAMES R. HOLTON and H. THOMAS ROSSBY A complete list of books in this series appears at the end of this volume.

Introduction to Micrometeorology Second Edition

S. Pal Arya

DEPARTMENT OF MARINE, EARTH AND ATMOSPHERIC SCIENCES NORTH CAROLINA STATE UNIVERSITY RALEIGH, NORTH CAROLINA, USA

San Diego San Francisco New York Boston London Sydney Tokyo

This book is printed on acid-free paper. Copyright # 2001, 1998 by ACADEMIC PRESS All Rights Reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Requests for permission to make copies of any part of the work should be mailed to the following address: Permissions Department, Harcourt, Inc., 6277 Sea Harbor Drive, Orlando, Florida, 32887-6777. Academic Press A Harcourt Science and Technology Company 525 B Street, Suite 1900, San Diego, California 92101-4495, USA http://www.academicpress.com Academic Press A Harcourt Science and Technology Company Harcourt Place, 32 Jamestown Road, London NW1 7BY, UK http://www.academicpress.com ISBN 0-12-059354-8

Typeset by Paston PrePress Ltd, Beccles, Su€olk Printed and bound in Great Britain by MPG, Bodmin, Cornwall 01 02 03 04 05 06 MP 9

8 7 6 5

4 3 2 1

To my wife, Nirmal, and my children, Niki, Sumi and Vishal

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Contents

Preface to Second Edition Preface to First Edition Symbols

xii xiii xvii

Chapter 1 Introduction

1

1.1 1.2 1.3

1 5 6 9

Scope of Micrometeorology Micrometeorology versus Microclimatology Importance and Applications of Micrometeorology Problems and Exercises

Chapter 2 Energy Budget Near the Surface

11

2.1 2.2 2.3 2.4 2.5 2.6

11 12 19 20 25 25 26

Energy Fluxes at an Ideal Surface Energy Balance Equations Energy Budgets of Bare Surfaces Energy Budgets of Canopies Energy Budgets of Water Surfaces Applications Problems and Exercises

Chapter 3 Radiation Balance Near the Surface

28

3.1 3.2 3.3 3.4 3.5 3.6

28 32 36 38 39 41

Radiation Laws and De®nitions Shortwave Radiation Longwave Radiation Radiation Balance near the Surface Observations of Radiation Balance Radiative Flux Divergence vii

viii

Contents

3.7

Applications Problems and Exercises

43 44

Chapter 4 Soil Temperatures and Heat Transfer

46

4.1 4.2 4.3 4.4 4.5 4.6 4.7

46 47 48 51 52 56 60 60

Surface Temperature Subsurface Temperatures Thermal Properties of Soils Theory of Soil Heat Transfer Thermal Wave Propagation in Soils Measurement and Parameterization of Ground Heat Flux Applications Problems and Exercises

Chapter 5 Air Temperature and Humidity in the PBL

62

5.1 5.2 5.3 5.4 5.5 5.6 5.7

62 63 71 77 78 82 85 86

Factors In¯uencing Air Temperature and Humidity Basic Thermodynamic Relations and the Energy Equation Local and Nonlocal Concepts of Static Stability Mixed Layers, Mixing Layers and Inversions Vertical Temperature and Humidity Pro®les Diurnal Variations Applications Problems and Exercises

Chapter 6 Wind Distribution in the PBL 6.1 6.2 6.3 6.4 6.5 6.6 6.7

Factors In¯uencing Wind Distribution Geostrophic and Thermal Winds Friction E€ects on the Balance of Forces Stability E€ects on the PBL Winds Observed Wind Pro®les Diurnal Variations Applications Problems and Exercises

89 89 89 95 99 101 107 109 110

Chapter 7 An Introduction to Viscous Flows

113

7.1 7.2

113 116

Inviscid and Viscous Flows Laminar and Turbulent Flows

Contents 7.3 7.4 7.5 7.6 7.7 7.8

Navier±Stokes Equations of Motion Laminar Plane-Parallel Flows Laminar Ekman Layers Developing Laminar Boundary Layers Heat Transfer in Laminar Flows Applications Problems and Exercises

ix 117 119 124 132 136 139 139

Chapter 8 Fundamentals of Turbulence

142

8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8

142 146 147 149 152 155 156 160 161

Instability of Flow and Transition to Turbulence The Generation and Maintenance of Turbulence General Characteristics of Turbulence Mean and Fluctuating Variables Variances and Turbulent Fluxes Eddies and Scales of Motion Fundamental Concepts, Hypotheses, and Theories Applications Problems and Exercises

Chapter 9 Models and Theories of Turbulence

163

9.1 9.2 9.3 9.4

163 171 177 185 185

Mathematical Models of Turbulent Flows Gradient-transport Theories Dimensional Analysis and Similarity Theories Applications Problems and Exercises

Chapter 10 Near-neutral Boundary Layers

188

10.1 10.2 10.3 10.4 10.5

Velocity-pro®le Laws Surface Roughness Parameters Surface Drag and Resistance Laws Turbulence Applications Problems and Exercises

188 197 203 208 211 211

Chapter 11 Thermally Strati®ed Surface Layer

213

11.1 The Monin±Obukhov Similarity Theory

213

x 11.2 11.3 11.4 11.5 11.6

Contents Empirical Forms of Similarity Functions Wind and Temperature Pro®les Drag and Heat Transfer Relations Methods of Determining Momentum and Heat Fluxes Applications Problems and Exercises

217 223 224 226 239 239

Chapter 12 Evaporation from Homogeneous Surfaces

242

12.1 12.2 12.3 12.4 12.5

242 243 245 248 264 264

The Process of Evaporation Potential Evaporation and Evapotranspiration Modi®ed Monin±Obukhov Similarity Relations Micrometeorological Methods of Determining Evaporation Applications Problems and Exercises

Chapter 13 Strati®ed Atmospheric Boundary Layers

266

13.1 13.2 13.3 13.4 13.5

266 270 277 310 317 317

Types of Atmospheric Boundary Layers Similarity Theories of the PBL Mathematical Models of the PBL Parameterization of the PBL Applications Problems and Exercises

Chapter 14 Nonhomogeneous Boundary Layers

321

14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8

321 322 328 334 336 346 353 362 362

Types of Surface Inhomogeneities Step Changes in Surface Roughness Step Changes in Surface Temperature Air Modi®cations over Water Surfaces Air Modi®cations over Urban Areas Building Wakes and Street Canyon E€ects Other Topographical E€ects Applications Problems and Exercises

Chapter 15 Agricultural and Forest Micrometeorology

365

15.1 Flux-pro®le Relations above Plant Canopies 15.2 Radiation Balance within Plant Canopies

365 370

Contents 15.3 15.4 15.5 15.6

Wind Distribution in Plant Canopies Temperature and Moisture Fields Turbulence in and above Plant Canopies Applications Problems and Exercises

References Index

xi 375 378 381 388 389 391 405

Preface to the Second Edition

It has been nearly twelve years since the publication of the ®rst edition of this book. During this time, many journal papers have been published on various aspects of micrometeorology and the related areas of atmospheric boundary layer and turbulence. Several new books and monographs have also been published (Stull, 1988; Sorbjan, 1989; Monteith and Unsworth, 1990; Garratt 1992; Kaimal and Finnigan, 1994). The primary motivation for preparing this second edition came from the perceived need to update the material, incorporate some worked-out example problems in each chapter, and include a comprehensive treatment of observations, theories and models of the atmospheric boundary layer. I hope that I have accomplished these tasks, while retaining the original scope and objectives given in my Preface to the First Edition. Updating has resulted in an expanded list of references. Worked-out example problems should help the reader in solving other problems and exercises given at the end of each chapter. The new Chapter 13 provides a comprehensive review of strati®ed atmospheric boundary layers, while much of the material of the old chapter on the marine atmospheric boundary layer has been incorporated in other chapters. Other chapters have also been revised and updated, wherever appropriate. I would like to thank the reviewers of my proposal for the second edition for their excellent suggestions. I would also like to thank the publishers and authors for their permission to reproduce many ®gures and tables in the book. The original sources are acknowledged in the legends of ®gures and tables, whenever applicable. Finally, I wish to acknowledge the help of Ms Mel Defeo and Ms Michele Marcusky for word-processing of the manuscript and in performing the painstaking tasks of spellchecking, proofreading, and correcting several times during the last one and a half years.

xii

Preface to the First Edition

Many university departments of meteorology or atmospheric sciences o€er a course in micrometeorology as a part of their undergraduate curricula. Some graduate programs also include an introductory course in micrometeorology or environmental ¯uid mechanics, in addition to more advanced courses on planetary boundary layer (PBL) and turbulence. While there are a number of excellent textbooks and monographs available for advanced graduate courses, there is no suitable text for an introductory undergraduate or graduate course in micrometeorology. My colleagues and I have faced this problem for more than ten years, and we suspect that other instructors of introductory micrometeorology courses have faced the same problem ± lack of a suitable textbook. At ®rst, we tried to use R.E. Munn's `Descriptive Micrometeorology,' but soon found it to be much out-of-date. There has been almost an explosive development of the ®eld during the past two decades, as a result of increased interest in micrometeorological problems and advances in computational and observational facilities. When Academic Press approached me to write a textbook, I immediately saw an opportunity for ®lling a void and satisfying an acutely felt need for an introductory text, particularly for undergraduate students and instructors. I also had in mind some incoming graduate students having no background in micrometeorology or ¯uid mechanics. Instructors may also ®nd the latter half of the book suitable for an introductory or ®rst graduate course in atmospheric boundary layer and turbulence. Finally, it may serve as an information source for boundary-layer meteorologists, air pollution meteorologists, agricultural and forest micrometeorologists, and environmental scientists and engineers. Keeping in mind my primary readership, I have tried to introduce the various topics at a suciently elementary level, starting from the basic thermodynamic and ¯uid dynamic laws and concepts. I have also given qualitative descriptions based on observations before introducing more complex theoretical concepts and quantitative relations. Mathematical treatment is deliberately kept simple, presuming only a minimal mathematical background of upper-division science majors. Uniform notation and symbols are used throughout the text, although certain symbols do represent di€erent things in xiii

xiv

Preface to the First Edition

di€erent contexts. The list of symbols should be helpful to the reader. Sample problems and exercises given at the end of each chapter should be useful to students, as well as to instructors. Many of these were given in homework assignments, tests, and examinations for our undergraduate course in micrometeorology. The book is organized in the form of ®fteen chapters, arranged in order of increasing complexity and what I considered to be a natural order of the topics covered in the book. The scope and importance of micrometeorology and turbulent exchange processes in the PBL are introduced in Chapter 1. The next three chapters describe the energy budget near the surface and its components, such as radiative, conductive, and convective heat ¯uxes. Chapter 5 reviews basic thermodynamic relations and presents typical temperature and humidity distributions in the PBL. Wind distribution in the PBL and simple dynamics, including the balance of forces on an air parcel in the PBL, are discussed in Chapter 6. The emphasis, thus far, is on observations, and very little theory is used in these early chapters. The viscous ¯ow theory, fundamentals of turbulence, and classical semiempirical theories of turbulence, which are widely used in micrometeorology and ¯uid mechanics, are introduced in Chapters 7, 8, and 9. Instructors of undergraduate courses may like to skip some of the material presented here, particularly the Reynolds-averaged equations for turbulent motion. Chapters 10±12 present the surface-layer similarity theory and micrometeorological methods and observations using the similarity framework. These three chapters would constitute the core of any course in micrometeorology. The last three chapters cover the more specialized topics of the marine atmospheric boundary layer, nonhomogeneous boundary layers, and micrometeorology of vegetated surfaces. Since these topics are still hotly pursued by researchers, the reader will ®nd much new information taken from recent journal articles. The instructor should be selective in what might be included in an undergraduate course. Chapters 7±15 should also be suitable for a ®rst graduate course in micrometeorology or PBL. Finally, I would like to acknowledge the contribution of my colleagues, Jerry Davis, Al Riordan, and Sethu Raman, and of my students for reviewing certain parts of the manuscript and pointing out many errors and omissions. I am most grateful to Professor Robert Fleagle for his thorough and timely review of each chapter, without the bene®t of the ®gures and illustrations. His comments and suggestions were extremely helpful in preparing the ®nal draft. I would also like to thank the publishers and authors of many journal articles, books, and monographs for their permission to reproduce many ®gures and tables in this book. The original sources are acknowledged in ®gure legends, whenever applicable. Still, many ideas, explanations, and discussions in the text are presented without speci®c acknowledgments of the original sources, because, in many cases, the original sources were not known and also because, I thought,

Preface to the First Edition

xv

giving too many references in the text would actually distract the reader. In the end, I wish to acknowledge the help of Brenda Batts and Dava House for typing the manuscript and Pat Bowers and LuAnn Salzillo for drafting ®gures and illustrations.

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Symbols

A A A As Ay a a a ab ai ay a1 a2 B B B b C C C CD Cd CDN Cg CH CHN CS Cs CW

Amplitude of thermal wave in the subsurface medium (Chapter 4) Aerodynamic surface area of vegetation per unit volume (Chapter 15) Similarity constant or function (Chapters 10, 13) Amplitude of thermal wave at the surface Mean advection terms in the thermodynamic energy equation Albedo or shortwave re¯ectivity of the surface (Chapter 3) Inverse length scale (Chapter 7) An empirical constant (Chapter 10) Acceleration due to buoyancy Empirical constant Instantaneous advection terms in the thermodynamic energy equation (Chapter 9) Empirical constant Empirical constant Bowen ratio Buoyancy production term in the turbulent kinetic energy equation (Chapter 8) Similarity constant or function (Chapters 10, 13) Boltzmann constant Heat capacity Empirical constant (Chapter 9) Coriolis force (Chapter 6) Surface drag coecient Drag coecient (Chapter 15) Drag coecient in neutral stability Heat capacity of the near-ground soil layer (Chapter 4) Heat transfer coecient Heat transfer coecient in neutral stability Smagorinski constant (Chapter 13) Volumetric heat capacity of soil (Chapter 4) Water vapor transfer coecient xvii

xviii Cw Cy C0 c c c cg cm cm cp cv D D D/Dt d d d dH dm dP dT d0 E E Ea Ep E0 e e er es e0 F F F Fc FL Fu Fv f f

Symbols A constant (Chapter 9) An empirical constant Phase speed of dominant waves Speci®c heat Speed of light (electromagnetic waves) in vacuum (Chapter 3) Empirical constant (Chapter 10) Geostrophic drag coecient Electromagnetic wave speed in a medium (Chapter 3) A constant (Chapter 14) Speci®c heat at constant pressure Speci®c heat at constant volume Reference depth in the sub-medium (Chapter 4) Viscous dissipation term in the turbulent kinetic energy equation (Chapter 8) Total derivative Distance of the earth from the sun (Chapter 3) Damping depth of thermal wave in the submedium (Chapter 4) Dimensionless constant (Chapter 13) Heat added to a parcel Mean distance between the earth and the sun Change in the air pressure Change in the air temperature Zero-plane displacement Rate of evaporation or condensation Average turbulence kinetic energy Drying power of air Potential evaporation or evapotranspiration Rate of evaporation or condensation at the surface Water vapor pressure (Chapters 5 and 12) Fluctuating turbulence kinetic energy (Chapters 9 and 13) Saturated vapor pressure at reference height (Chapter 12) Water vapor pressure at saturation (Chapter 5) Water vapor pressure at the surface or at z = z0 Function of certain variables (Chapter 9) Froude number (Chapters 13 and 14) Friction force (Chapter 6) Critical Froude number Froude number based on length of the hill A similarity function A similarity function Coriolis parameter Function of variables (Chapter 9)

Symbols f' G G Gr G0 G0 g H H H1 HF HG Hg Hi Hin HL Hout Hs H0 h h hE hi hi hp h0 I I0 i iu iv iw K K KE Kh Km Kmr Kw K k

Normalized velocity as a function of Z Magnitude of geostrophic wind Geostrophic wind vector Grashof number Magnitude of surface geostrophic wind Surface geostrophic wind vector Acceleration due to gravity Sensible (direct) heat ¯ux Building height (Chapter 14) Soil heat ¯ux at depth z1 Anthropogenic heat ¯ux in an urban area Ground heat ¯ux to or from the subsurface medium Soil heat ¯ux at depth dg Heat ¯ux at the inversion base Energy coming in Latent heat ¯ux Energy going out Height of the dividing streamline Sensible heat ¯ux at the surface Boundary layer thickness Depth of channel ¯ow or distance between two parallel planes (Chapter 7) Ekman depth Height of surface inversion (Chapter 5) Internal boundary layer thickness (Chapter 14) Planck's constant Average height of roughness elements or plant canopy Insolation at the surface Insolation at the toppof the  atmosphere Imaginary number 1 Longitudinal turbulence intensity Lateral turbulence intensity Vertical turbulence intensity E€ective viscosity (Chapter 7) Subgrid-scale eddy viscosity (Chapter 13) Di€usivity of turbulence kinetic energy Eddy di€usivity of heat Eddy viscosity or di€usivity of momentum Eddy viscosity at the reference height Eddy di€usivity of water vapor Dimensionless eddy viscosity (Chapter 13) Von Karman constant

xix

xx k km L L LAI Le l l lb lh lm lo ` `e M Mb Md Mw M0 m m m md mw N Nu n n P P P PAI Pe Pr P0 p p0 p1 Q Qr Qs Q0

Symbols Thermal conductivity (Chapter 4) Hydraulic conductivity of the subsurface medium Obukhov length A characteristic length scale (Chapter 9) Leaf area index Latent heat of evaporation/condensation Large eddy length scale (Chapter 8) Fluctuating mixing length (Chapter 9) Buoyancy-limited mixing length (Chapter 13) Mean mixing length of heat Mean mixing length of momentum Mixing length in the outer layer (Chapter 13) Large-eddy length scale (Chapter 13) Dissipation length scale (Chapter 13) Soil moisture ¯ow rate Soil moisture ¯ow rate at the bottom of layer Mass of dry air Mass of water vapor Soil moisture ¯ow rate at the surface Mean molecular mass of air (Chapter 5) Exponent in the power-law wind pro®le equation (Chapter 10) A coecient (Chapter 15) Mean molecular mass of dry air Mean molecular mass of water vapor Brunt±Vaisala frequency Nusselt number Exponent in the power-law eddy viscosity pro®le (Chapter 10) A coecient (Chapter 15) Mean air pressure Period of the wave (Chapter 4) Pressure-gradient force Plant area index Peclet number Prandtl number Pressure of the reference atmosphere Instantaneous pressure Pressure of the reference state Deviation of pressure from the reference state Mean speci®c humidity Saturation speci®c humidity at reference height (Chapter 12) Mean speci®c humidity at saturation (Chapter 5) Speci®c humidity at z = z0

Symbols q q R R Ra RD Rd Re Rf Ri RiB Ric RL RL; RL: RN RO RS RS; RS: Rs R Rl R0 r rH rM rw S S S S SF SH S0 s s T T Tc Teb

Speci®c humidity ¯uctuation Speci®c humidity scale for the surface layer Radiative energy ¯ux or radiation (Chapter 3) Speci®c gas constant (Chapter 5) Rayleigh number Di€use radiation Speci®c gas constant for dry air Reynolds number Flux Richardson number Gradient Richardson number Bulk Richardson number Critical Richardson number Longwave radiation (¯ux) Incoming (downward) longwave radiation Outgoing (upward) longwave radiation Net radiation Surface Rossby number Shortwave radiation (¯ux) Incoming (downward) shortwave radiation Outgoing (upward) shortwave radiation Solar radiation ¯ux at the surface Absolute gas constant Radiative energy ¯ux density per unit wavelength Solar radiation ¯ux at the top of the atmosphere Correlation coecient Aerial resistance to heat transfer Resistance to momentum transfer Resistance to water vapor transfer Shear production term in the turbulent kinetic energy equation (Chapter 8) Soil moisture content (Chapter 12) Skewness (Chapter 13) Mean value of a passive scalar Speed-up factor Source or sink of heat Solar constant Static stability Fluctuating passive scalar Mean air temperature Temperature of the subsurface medium (Chapter 4) Complex stress variable Equivalent blackbody temperature

xxi

xxii Tg Tm Tm Tr Ts Ts Tu Tv Tvp Tv0 T Tx Ty T0 T1 T10 t U Ug Uh Uh Um Ur U? U0 U10 u u~ uf u` u V V V V Vg Vm Vm v v~ W W

Symbols Temperature of the near-ground soil layer Mean temperature of the surface and submedium Mean temperature of the lower soil layer Turbulent transport of turbulent kinetic energy Surface temperature (Chapter 4) Sea-surface temperature (Chapter 12) Urban air temperature Mean virtual temperature of moist air Mean virtual temperature of the parcel Virtual temperature at the reference state Convective temperature scale Normalized shear stress in x direction Normalized shear stress in y direction Air temperature at the reference state Deviation in the temperature from the reference state Air temperature at 10 m above the surface Time Mean velocity component in x direction Geostrophic wind component in x direction Velocity at z = h Velocity of the moving surface (Chapter 7) Mixed-layer averaged wind component in x direction Wind speed at the reference height Ambient wind speed Mean velocity in the approach ¯ow Wind speed at a reference height of 10 m Fluctuating velocity component in x direction Instantaneous velocity in x direction Local free-convective velocity scale Characteristic velocity scale of turbulence (Chapter 8) Friction velocity Mean velocity component in y direction Volume (Chapter 2) Characteristic velocity scale (Chapter 9) Wind vector Geostrophic wind component in y direction Layer-averaged wind speed in the PBL Mixed-layer averaged wind component in y direction Fluctuating velocity component in y direction Instantaneous velocity in y direction Mean velocity component in z (vertical) direction Building width (Chapter 14)

Symbols WAI Wh W w w~ we z zi zm zr z' zp z0 z01 z02 a a ah am aw al ay a0 b b G Gs g g g D D DA DHs DM DTs DTu7r DU DU

xxiii

Woody-element area index Mean vertical motion at the top of the PBL Convective velocity scale Fluctuating velocity component in z (vertical) direction Instantaneous velocity in z direction Entrainment velocity Height above or depth below the surface or an appropriate reference plane Height of inversion base above the surface Geometric mean of the two heights z1 and z2 Reference height Height above the ground level Height of a constant pressure surface Roughness length parameter Roughness parameter of the upstream surface Roughness parameter of the downstream surface Cross-isobar angle (Chapter 6) Empirical constant (Chapter 15) Molecular di€usivity of heat or thermal di€usivity Soil moisture di€usivity Molecular di€usivity of water vapor in air Absorptivity at wavelength l Dimensionless coecient (Chapter 12) Cross-isobar angle at the surface Coecient of thermal expansion Slope angle of an inclined plane (Chapter 7) Adiabatic lapse rate Saturated adiabatic lapse rate Solar zenith angle (Chapter 3) Psychrometer constant (Chapter 12) Potential temperature gradient above the PBL (Chapter 13) Gradient operator Slope of the saturation vapor pressure versus temperature curve (Chapter 12) Elemental area Rate of energy storage Rate of storage of soil moisture Di€erence between the maximum and minimum surface temperatures Di€erence between the urban and rural air temperatures Di€erence in mean velocities at two heights Velocity de®cit in the wake

xxiv DUg DVg DY Dx Dy Dz Dz H2 e e el z Z Z Z Y Ym Yr Yu Yv Y0 Y01 Y02 y ~ yf  k l l lmax m mg  n x xt P p r rp r0

Symbols Change in Ug over a layer Change in Vg over a layer Di€erence between potential temperatures at two heights Small increment in x Small increment in y Small increment in z z2 7 z 1 Laplacian operator Overall infrared emissivity (Chapter 3) Rate of energy dissipation (Chapters 8, 9, 13) Emissivity at wavelength l Monin±Obukhov stability parameter Normalized distance from the surface (Chapter 7) Sea-surface elevation (Chapter 13) Kolmogorov's microscale of length (Chapter 8) Mean potential temperature Mixed-layer averaged potential temperature Rural air potential temperature Urban air potential temperature Mean virtual potential temperature Potential temperature at z = z0 Temperature of the upstream surface Temperature of the downstream surface Fluctuating potential temperature Instantaneous potential temperature Local free convective temperature scale Friction temperature scale Wave number Wavelength (Chapter 3) Frontal area density of roughness elements (Chapter 10) Wavelength corresponding to spectral maximum Dynamic viscosity Heat transfer coecient Dimensionless stability parameter (Chapter 13) Kinematic viscosity Dimensionless height parameter (Chapters 10, 13) Dimensionless PBL height Dimensionless group The ratio of circumference to diameter of a circle Mass density of air or other medium Mass density of the air parcel Density of the reference state

Symbols r1 s sq su sv sw sy t t t0 fh fm fw ch cm cw O V o

Deviation in the density from the reference state Stefan±Boltzmann constant Standard deviation of speci®c humidity ¯uctuations Standard deviation of velocity ¯uctuations in x direction Standard deviation of velocity ¯uctuations in y direction Standard deviation of velocity ¯uctuations in z direction Standard deviation of temperature ¯uctuations Shearing stress Shear stress vector Surface shear stress or drag Dimensionless potential temperature gradient Dimensionless wind shear Dimensionless speci®c humidity gradient M±O similarity function for normalized potential temperature M±O similarity function for normalized velocity M±O similarity function for normalized speci®c humidity Rotational speed of the earth Earth's rotational velocity vector Wave frequency

xxv

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Chapter 1

Introduction

1.1 Scope of Micrometeorology Atmospheric motions are characterized by a variety of scales ranging from the order of a millimeter to as large as the circumference of the earth in the horizontal direction and the entire depth of the atmosphere in the vertical direction. The corresponding time scales range from a tiny fraction of a second to several months or years. These scales of motions are generally classi®ed into three broad categories, namely, micro-, meso-, and macroscales. Sometimes, terms such as local, regional, and global are used to characterize the atmospheric scales and the phenomena associated with them. Micrometeorology is a branch of meteorology which deals with the atmospheric phenomena and processes at the lower end of the spectrum of atmospheric scales, which are variously characterized as microscale, small-scale, or local-scale processes. The scope of micrometeorology is further limited to only those phenomena which originate in and are dominated by the shallow layer of frictional in¯uence adjoining the earth's surface, commonly known as the atmospheric boundary layer (ABL) or the planetary boundary layer (PBL). Thus, some of the small-scale phenomena, such as convective clouds and tornadoes, are considered outside the scope of micrometeorology, because their dynamics is largely governed by mesoscale and macroscale weather systems. In particular, micrometeorology deals with the exchanges of heat (energy), mass, and momentum occurring continuously between the atmosphere and the earth's surface, including the subsurface medium. The energy budget at or near the surface on a short-term (say, hourly) basis is an important aspect of the di€erent types of energy exchanges involved in the earth±atmosphere±sun system. Vertical distributions of meteorological variables such as wind, temperature, and humidity, as well as trace gas concentrations and their role in the energy balance near the surface, also come under the scope of micrometeorology. In addition to the short-term averaged values of meteorological variables, more or less random ¯uctuations of the same in time and space around their respective average values are of considerable interest. The statistics of these socalled turbulent ¯uctuations are intimately related to the above-mentioned 1

2

1 Introduction

exchange processes and, hence, constitute an integral part of micrometeorology or boundary-layer meteorology. 1.1.1 Atmospheric boundary layer A boundary layer is de®ned as the layer of a ¯uid (liquid or gas) in the immediate vicinity of a material surface in which signi®cant exchange of momentum, heat, or mass takes place between the surface and the ¯uid. Sharp variations in the properties of the ¯ow, such as velocity, temperature, and mass concentration, also occur in the boundary layer. The atmospheric boundary layer is formed as a consequence of the interactions between the atmosphere and the underlying surface (land or water) over time scales of a few hours to about 1 day. Over longer periods the earth± atmosphere interactions may span the whole depth of the troposphere, typically 10 km, although the PBL still plays an important part in these interactions. The in¯uence of surface friction, heating, etc., is quickly and eciently transmitted to the entire PBL through the mechanism of turbulent transfer or mixing. Momentum, heat, and mass can also be transferred downward through the PBL to the surface by the same mechanism. A schematic of the PBL, as the lower part of the troposphere, over an underlying rough surface is given in Figure 1.1. Also depicted in the same ®gure is the frequently used division of the atmospheric boundary layer into a surface layer and an outer or upper layer. The vertical dimensions (heights) given in Figure 1.1 are more typical of the near-neutral stability observed during strong winds and overcast skies; these are highly variable in both time and space.

Figure 1.1 Schematic of the planetary boundary layer as the lower part of the troposphere. [From Arya (1982).]

1.1 Scope of Micrometeorology

3

The atmospheric PBL thickness over land surfaces varies over a wide range (several tens of meters to several kilometers) and depends on the rate of heating or cooling of the surface, strength of winds, the roughness and topographical characteristics of the surface, large-scale vertical motion, horizontal advections of heat and moisture, and other factors. In the air pollution literature the PBL height is commonly referred to as the mixing height or depth, since it represents the depth of the layer through which pollutants released from the near-surface sources are eventually mixed. As a result, the PBL is generally dirtier than the free atmosphere above it. The contrast between the two is usually quite sharp over large cities and can be observed from an aircraft as it leaves or enters the PBL. In response to the strong diurnal cycle of heating and cooling of land surfaces during fair-weather conditions, the PBL thickness and other boundary layer characteristics also display strong diurnal variations. For example, the PBL height over a dry land surface in summer can vary from less than 100 m in the early morning to several kilometers in the late afternoon. Following sunrise on a clear day, the continuous heating of the land surface by the sun and the resulting thermal mixing in the PBL cause the PBL depth to increase steadily throughout the day and attain a maximum value of the order of 1 km (range ^0.2±5 km) in the late afternoon. Later in the evening and throughout the night, on the other hand, the radiative cooling of the ground surface results in the suppression or weakening of turbulent mixing and consequently in the shrinking of the PBL depth to a typical value of the order of only 100 m (range ^20±500 m). Thus the PBL depth waxes and wanes in response to the diurnal heating and cooling cycle. The winds, temperatures, and other properties of the PBL may also be expected to exhibit strong diurnal variations. Diurnal variations of the PBL height and other meteorological variables are found to be much smaller over large lakes, seas and oceans, because of the small diurnal changes of the water surface temperature due to the large heat capacity of the mixed layer in water. Other temporal variations of the PBL height and structure often occur as a result of the development and the passage of mesoscale and synoptic systems. Generally, the PBL becomes thinner under the in¯uence of large-scale subsidence (downward motion) and the low-level horizontal divergence associated with a high-pressure system (anticyclone). On the other hand, the PBL can grow to be very deep and merge with towering clouds in disturbed weather conditions that are associated with low-pressure systems (cyclones). It is often dicult to distinguish the PBL top from in-cloud circulations under these conditions; the cloud base is generally used as an arbitrary cuto€ for the boundary layer top. Spatial variations of the PBL depth and structure occur as a result of changes in land use and topography of the underlying surface. Spatial variations of meteorological variables in¯uenced by mesoscale and large-scale systems also

4

1 Introduction

lead to similar variations in the boundary layer. On a ¯at and homogeneous surface, however, the PBL is generally considered horizontally homogeneous. 1.1.2 The surface layer Some investigators limit the scope of micrometeorology to only the so-called atmospheric surface layer, which comprises the lowest one-tenth or so of the PBL and in which the earth's rotational or Coriolis e€ects can be ignored. Such a restriction may not be desirable, because the surface layer is an integral part of and is much in¯uenced by the PBL as a whole, and the top of this layer is not physically as well de®ned as the top of the PBL. The latter represents the fairly sharp boundary between the irregular and almost chaotic (turbulent) motions in the PBL and the smooth and streamlined (nonturbulent) ¯ow in the free atmosphere above. The PBL top can easily be detected by ground-based remote-sensing devices, such as acoustic sounder, lidar, etc., and can also be inferred from temperature, humidity, and wind soundings. However, the surface layer is more readily amenable to observations from the surface, as well as from micrometeorological masts and towers. It is also the layer in which most human beings, animals, and vegetation live and in which most human activities take place. The sharpest variations in meteorological variables with height occur within the surface layer and, consequently, the most signi®cant exchanges of momentum, heat, and mass also occur in this layer. Therefore, it is not surprising that the surface layer has received far greater attention from micrometeorologists and microclimatologists than has the outer part of the PBL. The lowest part of the surface layer in which the in¯uence of individual roughness elements can readily be discerned is called the roughness layer or the canopy layer (see Figure 1.1). For bare land surfaces, it is quite thin and often ignored. For grasslands and other vegetated surfaces, the height of the roughness layer is proportional to (say, 1.5 times) the average height of vegetation. In built-up (suburban and urban) areas, the height of roughness or canopy layer may depend on the spatial distribution and heights of buildings in the particular area. Over large city centers, the roughness layer may comprise a signi®cant portion of the urban boundary layer, especially at night. 1.1.3 Turbulence Turbulence refers to the apparently chaotic nature of many ¯ows, which is manifested in the form of irregular, almost random ¯uctuations in velocity, temperature, and scalar concentrations around their mean values in time and

1.2 Micrometeorology versus Microclimatology

5

space. Atmospheric turbulence is always manifested in the form of gustiness of winds, so that gustiness can be regarded as a simple measure of turbulence strength or intensity. The motions in the atmospheric boundary layer are almost always turbulent. In the surface layer, turbulence is more or less continuous, while it may be intermittent and patchy in the upper part of the PBL and is sometimes mixed with internal gravity waves. The PBL top is usually de®ned as the level where turbulence disappears or becomes insigni®cant. Near the surface, atmospheric turbulence manifests itself through the ¯utter of leaves of trees and blades of grass, swaying of branches of trees and plants, irregular movements of smoke and dust particles, generation of ripples and waves on water surfaces, and a variety of other visible phenomena. In the upper part of the PBL, turbulence is manifested by irregular motions of kites and balloons, spreading of smoke and other visible pollutants as they exit tall stacks or chimneys, and ¯uctuations in the temperature and refractive index encountered in the transmission of sound, light, and radio waves. 1.2 Micrometeorology versus Microclimatology The di€erence between micrometeorology and microclimatology is primarily in the time of averaging the variables, such as air velocity, temperature, humidity, etc. While the micrometeorologist is primarily interested in ¯uctuations, as well as in the short-term (of the order of an hour or less) averages of meteorological variables in the PBL or the surface layer, the microclimatologist mainly deals with the long-term (climatological) averages of the same variables. The latter is also interested in diurnal and seasonal variations, as well as in very long-term trends of meteorological parameters. In micrometeorology, detailed examination of small-scale temporal and spatial structure of ¯ow and thermodynamic variables is often necessary to gain a better understanding of the phenomena of interest. For example, in order to determine the short-time average concentrations on the ground some distance away from a pollutant source, one needs to know not only the mean winds that are responsible for the mean transport of the pollutant, but also the statistical properties of wind gusts (turbulent ¯uctuations) that are responsible for spreading or di€using the pollutant as it moves along the mean wind. In microclimatology, one deals with long-term averages of meteorological variables in the near-surface layer of the atmosphere. Details of ®ne structure are not considered so important, because their e€ects on the mean variables are smoothed out in the averaging process. Still, one cannot ignore the long-term consequences of the small-scale turbulent transfer and exchange processes. Despite the above-mentioned di€erences, micrometeorology and microclimatology have much in common, because they both deal with similar atmospheric

6

1 Introduction

processes occurring near the surface. Their interrelationship is further emphasized by the fact that long-term averages dealt with in the latter can, in principle, be obtained by the integration in time of the short-time averaged micrometeorological variables. It is not surprising, therefore, to ®nd some of the fundamentals of micrometeorology described in books on microclimatology (e.g., Geiger, 1965; Oke, 1987; Rosenberg et al., 1983; Geiger et al., 1995). Likewise, microclimatological information is found in and serves useful purposes in texts on micrometeorology (e.g., Sutton, 1953; Munn, 1966; Arya, 1988; Stull, 1988; Sorbjan, 1989; Garratt, 1992; Kaimal and Finnigan, 1994). 1.3 Importance and Applications of Micrometeorology Although the atmospheric boundary layer comprises only a tiny fraction of the atmosphere, the small-scale processes occurring within the PBL are useful to various human activities and are important for the well-being and even survival of life on earth. This is not merely because the air near the ground provides the necessary oxygen to human beings and animals, but also because this air is always in turbulent motion, which causes ecient mixing of pollutants and exchanges of heat, water vapor, etc., with the surface. 1.3.1 Turbulent transfer processes Turbulence is responsible for the ecient mixing and exchange of mass, heat, and momentum throughout the PBL. In particular, the surface layer turbulence is responsible for exchanging these properties between the atmosphere and the earth's surface. Without turbulence, such exchanges would have been at the molecular scale and minuscule in magnitudes (1073±1076 times the turbulent transfers that now commonly occur). Nearly all the energy which drives the large-scale weather and general circulation comes through the PBL. Through the ecient transfer of heat and moisture, the boundary layer turbulence moderates the microclimate near the ground and makes it habitable for animals, organisms, and plants. The atmosphere receives virtually all of its water vapor through turbulent exchanges near the surface. Evaporation from land and water surfaces is not only important in the surface water budget and the hydrological cycle, but the latent heat of evaporation is also an important component of the surface energy budget. This water vapor, when condensed on tiny dust particles and other aerosols (cloud condensation nuclei), leads to the formation of fog, haze, and clouds in the atmosphere. Besides water vapor, there are other important exchanges of mass within the PBL involving a variety of gases and particulates. Turbulence is important in

1.3 Importance and Applications of Micrometeorology

7

the exchange of carbon dioxide between plants and animals. Its atmospheric concentration has steadily been rising due to ever increasing use of fossil fuels in energy production and other industry, heating and cooking, and forest clearing by burning. This, in conjunction with similar increasing trends in the concentrations of methane and other radiatively active gases, threatens global climate warming. Many harmful toxic substances are also released into the atmosphere by human activities. Through ecient di€usion of the various pollutants released near the ground and mixing them throughout the PBL and parts of the lower troposphere, atmospheric turbulence prevents the fatal poisoning of life on earth. The quality of air we breathe depends to a large extent on the mixing capability of the PBL turbulence. The boundary layer turbulence also picks up pollen and other seeds of life, spreads them out, and deposits them over wide areas far removed from their origin. It lifts dust, salt particles, and other aerosols from the surface and spreads them throughout the lower atmosphere. Of these, the so-called cloud condensation nuclei are an essential ingredient in the condensation and precipitation processes in the atmosphere. Through the above-mentioned mass exchange processes between the earth's surface and the atmosphere, the radiation balance and the heat energy budget at or near the earth's surface are also signi®cantly a€ected. More direct e€ects of turbulent transfer on the surface heat energy budget are through sensible and latent heat exchanges between the surface and the atmosphere. Over land, the sensible heat exchange is usually more important than the latent heat of evaporation, but the reverse is true over large lakes and the oceans. Heat exchanges through the underlying bodies of water are also turbulent, especially those in the immediate vicinity of the surface. Turbulent transfer of momentum between the earth and atmosphere is also very important. It is essentially a one-way process in which the earth acts as a sink of atmospheric momentum (relative to earth). In other words, the earth's surface exerts frictional resistance to atmosphere motions and slows them down in the process. The moving air near the surface may be considered to exert an equivalent drag force on the surface. The rougher the surface, the larger would be the drag force per unit area of the surface. Some commonly encountered examples of this are the marked slowdown of surface winds in going from a large lake, bay, or sea to inland areas and from rural to urban areas. Perhaps the most vivid manifestations of the e€ect of increased surface drag are the rapid weakening and subsequent demise of hurricanes and other tropical storms as they move inland. Another factor responsible for this phenomenon is the marked reduction or cuto€ of the available latent heat energy to the storm from the surface. Over large lakes and oceans, wind drag is responsible for the generation of waves and currents in water, as well as for the movement of sea ice. In coastal areas, wind drag causes storm surges and beach erosion. Over land, drag exerts

8

1 Introduction

wind loads on vehicles, buildings, towers, bridges, cables, and other structures. Wind stress over sand surfaces raises dust and creates ripples and sand dunes. When accompanied by strong winds, the surface layer turbulence can be quite discomforting and even harmful to people, animals, and vegetation. Turbulent exchange processes in the PBL have profound e€ects on the evolution of local weather. Boundary layer friction is primarily responsible for the low-level convergence and divergence of ¯ow in the regions of lows and highs in surface pressures, respectively. The frictional convergence in a moist boundary layer is also responsible for the low-level convergence of moisture in low-pressure regions. The kinetic energy of the atmosphere is continuously dissipated by small-scale turbulence in the atmosphere. Almost one-half of this loss on an annual basis occurs within the PBL, even though the PBL comprises only a tiny fraction (less than 2%) of the total kinetic energy of the atmosphere. 1.3.2 Applications In the following text, we have listed some of the possible areas of application of micrometeorology together with the subareas or activities in which micrometeorological information may be especially useful. 1. Air pollution meteorology . Atmospheric transport and di€usion of pollutants . Atmospheric deposition on land and water surfaces . Prediction of local, urban, and regional air quality . Selection of sites for power plants and other major industries . Selection of sites for monitoring urban and regional air pollution . Industrial operations with emissions dependent on meteorological conditions . Agricultural operations such as dusting, spraying, and burning . Military operations with considerations of obscurity and dispersion of contaminants 2. Mesoscale meteorology . Urban boundary layer and heat island . Land±sea breezes . Drainage and mountain valley winds . Dust devils, water spouts, and tornadoes . Development of fronts and cyclones 3. Macrometeorology . Atmospheric predictability . Long-range weather forecasting

Problems and Exercises

9

. Siting and exposure of meteorological stations . General circulation and climate modeling 4. Agricultural and forest meteorology . Prediction of surface temperatures and frost conditions . Soil temperature and moisture . Evapotranspiration and water budget . Energy balance of a plant cover . Carbon dioxide exchanges within the plant canopy . Temperature, humidity, and winds in the canopy . Protection of crops and shrubs from strong winds and frost . Wind erosion of soil and protective measures . E€ects of acid rain and other pollutants on plants and trees 5. Urban planning and management . Prediction and abatement of ground fogs . Heating and cooling requirements . Wind loading and designing of structures . Wind sheltering and protective measures . Instituting air pollution control measures . Flow and dispersion around buildings . Prediction of road surface temperature and possible icing 6. Physical oceanography . Prediction of storm surges . Prediction of the sea state . Dynamics of the oceanic mixed layer . Movement of sea ice . Modeling large-scale oceanic circulations . Navigation . Radio transmission

Problems and Exercises 1. Compare and contrast the following terms: (a) microscale and macroscale atmospheric processes and phenomena; (b) micrometeorology and microclimatology; (c) atmospheric boundary layer and the surface layer. 2. On a schematic of the lower atmosphere, including the tropopause, indicate the PBL and the surface layer during typical fair-weather (a) daytime convective and (b) nighttime stable conditions.

10

1 Introduction

3. Discuss the following exchange processes between the earth and the atmosphere and the importance of boundary layer turbulence on them: (a) sensible heat; (b) water vapor; (c) momentum.

Chapter 2

Energy Budget Near the Surface

2.1 Energy Fluxes at an Ideal Surface The ¯ux of a property in a given direction is de®ned as its amount per unit time passing through a unit area normal to that direction. In this chapter, we will be concerned with ¯uxes of the various forms of heat energy at or near the surface. The SI units of energy ¯ux are J s71 m72 or W m72. The `ideal' surface considered here is relatively smooth, horizontal, homogeneous, extensive, and opaque to radiation. The energy budget of such a surface is considerably simpli®ed in that only the vertical ¯uxes of energy need to be considered. There are essentially four types of energy ¯uxes at an ideal surface, namely, the net radiation to or from the surface, the sensible (direct) and latent (indirect) heat ¯uxes to or from the atmosphere, and the heat ¯ux into or out of the submedium (soil or water). The net radiative ¯ux is a result of the radiation balance at the surface, which will be discussed in more detail in Chapter 3. During the daytime, it is usually dominated by the solar radiation and is almost always directed toward the surface, while at night the net radiation is much weaker and directed away from the surface. As a result, the surface warms up during the daytime, while it cools during the evening and night hours, especially under clear sky and undisturbed weather conditions. The direct or sensible heat ¯ux at and above the surface arises as a result of the di€erence in the temperatures of the surface and the air above. Actually, the temperature in the atmospheric surface layer varies continuously with height, with the magnitude of the vertical temperature gradient usually decreasing with height. In the immediate vicinity of the interface and within the so-called molecular sublayer, the primary mode of heat transfer in air is conduction, similar to that in solids. At distances beyond a few millimeters (the thickness of the molecular sublayer) from the interface, however, the primary mode of heat exchange becomes advection or convection involving air motions. The sensible heat ¯ux is usually directed away from the surface during the daytime hours, when the surface is warmer than the air above, and vice versa during the evening 11

12

2 Energy Budget Near the Surface

and nighttime periods. Thus, the heat ¯ux is down the average temperature gradient. The latent heat or water vapor ¯ux is a result of evaporation, evapotranspiration, or condensation at the surface and is given by the product of the latent heat of evaporation or condensation and the rate of evaporation or condensation. Evaporation occurs from water surfaces as well as from moist soil and vegetative surfaces, whenever the air above is drier, i.e., it has lower speci®c humidity than the air in the immediate vicinity of the surface and its transpirating elements (e.g., leaves). This is usually the situation during the daytime. On the other hand, condensation in the form of dew may occur on relatively colder surfaces at nighttime. The water vapor transfer through air does not involve any real heat exchange, except where phase changes between liquid water and vapor actually take place. Nevertheless, evaporation results in some cooling of the surface, which in the surface energy budget is represented by the latent heat ¯ux from the surface to the air above. The ratio of the sensible heat ¯ux to the latent heat ¯ux is called the Bowen ratio. The heat exchange through the ground medium is primarily due to conduction if the medium is soil, rock, or concrete. Through water, however, heat is transferred in the same way as it is through air, ®rst by conduction in the top few millimeters (molecular sublayer) from the surface and then by advection or convection in the deeper layers (surface layer, mixed layer, etc.) of water in motion. The depth of the submedium, which responds to and is a€ected by changes in the energy ¯uxes at the surface on a diurnal basis, is typically less than a meter for land surfaces and several tens of meters for lakes and oceans. 2.2 Energy Balance Equations 2.2.1 The surface energy budget For deriving a simpli®ed equation for the energy balance at an ideal surface, we assume the surface to be a very thin interface between the two media (air and soil or water), having no mass and heat capacity. The energy ¯uxes would ¯ow in and out of such a surface without any loss or gain due to the surface. Then, the principle of the conservation of energy at the surface can be expressed as R N = H + HL + HG

(2.1)

where RN is the net radiation, H and HL are the sensible and latent heat ¯uxes to or from the air, and HG is the ground heat ¯ux to or from the submedium. Here we use the sign convention that all the radiative ¯uxes directed toward the

2.2 Energy Balance Equations

13

surface are positive, while other (nonradiative) energy ¯uxes directed away from the surface are positive and vice versa. Equation (2.1) describes how the net radiation at the surface must be balanced by a combination of the sensible and latent heat ¯uxes to the air and the heat ¯ux to the subsurface medium. During the daytime, the surface receives radiative energy (RN 4 0), which is partitioned into sensible and latent heat ¯uxes to the atmosphere and the heat ¯ux to the submedium. Typically, H, HL, and HG are all positive over land surfaces during the day. This situation is schematically shown in Figure 2.1a. Actual magnitudes of the various components of the surface energy budget depend on many factors, such as the type of surface and its characteristics (soil moisture, texture, vegetation, etc.), geographical location, month or season, time of day, and weather. Under special circumstances, e.g., when irrigating a ®eld, H and/or HG may become negative and the latent heat ¯ux due to evaporative cooling of the surface may exceed the net radiation received at the surface. At night, the surface loses energy by outgoing radiation, especially during clear or partially overcast conditions. This loss is compensated by gains of heat from air and soil media and, at times, from the latent heat of condensation released during the process of dew formation. Thus, according to our sign

Figure 2.1 Schematic representation of typical surface energy budgets during (a) daytime and (b) nighttime.

14

2 Energy Budget Near the Surface

convention, all the terms of the surface energy balance [Equation (2.1)] for land surfaces are usually negative during the evening and nighttime periods. Their magnitudes are generally much smaller than the magnitudes of the daytime ¯uxes, except for HG. The magnitudes of HG do not di€er widely between day and night, although the direction or sign obviously reverses during the morning and evening transition periods, when other ¯uxes are also changing their signs (this does not happen simultaneously for all the ¯uxes, however). Figure 2.1b gives a schematic representation of the surface energy balance during the nighttime. From the above description of the diurnal cycle of the surface energy budget, it is clear that the ¯uxes of sensible and latent heat to or from the surface are governed by the diurnal cycle of net radiation. One can interpret Equation (2.1) in terms of the partitioning of net radiation into other ¯uxes (H, HL, and HG). The net radiation may be considered an external forcing, while the sensible, latent and ground heat ¯uxes are responses to this radiative forcing. Relative measures of this partitioning are the ratios H/RN, HL/RN, and HG/RN, which are expected to depend on the various surface, subsurface and meteorological characteristics. Diurnal variations of these ratios are expected to be much smaller than those of the individual ¯uxes. Of these, the ratio HG/RN might be expected to show the least variability, especially for a given land surface, because the above ratio may not be as sensitive to the variations in surface meteorological parameters as H/RN and HL/RN. In simpler parameterizations of the ground heat ¯ux, HG is assumed to be proportional to RN or related to RN through an empirical regression relationship (Doll et al., 1985). The ratio HG/RN is found to be larger for nighttime (RN 5 0) than for the daytime (RN 4 0) conditions. Values exceeding one have been observed during nighttime over urban surfaces, because of positive (upward) sensible heat ¯uxes even during nighttime. The simple relationship between the ground heat ¯ux and net radiation allows for the sum of the sensible and latent heat ¯uxes (H + HL) to be easily determined from the measurement or calculation of net radiation. Further separation of the total heat ¯ux into sensible and latent components can be made if the Bowen ratio, B = H/HL, can be estimated independently (e.g., using the Bowen ratio method to be discussed later). In terms of the Bowen ratio, one can obtain from Equation (2.1)

Hˆ

RN HG 1‡B 1

…2:2†

HL ˆ

R N HG 1‡B

…2:3†

2.2 Energy Balance Equations

15

The latent heat ¯ux can also be expressed as HL = LeE, where Le ^ 2.45 6 106 J kg71 is the latent heat of evaporation/condensation and E is the rate of evaporation/condensation. Example Problem 1 The average values of the ratio of ground heat ¯ux to net radiation determined from the Wangara experiment data are 0.30 and 0.52 for the daytime and nighttime, respectively. Assuming a Bowen ratio of 5.0, estimate the sensible and latent heat ¯uxes at that site when the measured net radiation is (a) 250 W m72, and (b) 755 W m72. Solution (a) RN = 250 W m72; HG = 0.30 6 250 = 75 W m72. From Equations (2.2) and (2.3), H = 145.8 W m72; HL = 29.2 W m72. Check that the above values satisfy Equation (2.1). (b) RN =755 W m72; HG = 70.52 6 55 = 728.6 W m72. From Equations (2.2) and (2.3), H = 722.0 W m72; HL = 74.4 W m72. Again, one can check that these values satisfy Equation (2.1). The energy budgets of extensive water surfaces (large lakes, seas, and oceans) di€er from those of land surfaces in several important ways. In the former, the combined value of HL and HG balances most of the net radiation, while H plays 5 1). Since the water surface temperature only a minor role (H 5 5 HL, or B 5 does not respond readily to solar heating due to the large heat capacity and depth of the subsurface mixed layer of a large lake or ocean, the air±water exchanges (H and HL) do not undergo large diurnal variations. An important factor to be considered in the energy balance over water surfaces is the penetration of solar radiation to depths of tens of meters. Radiation processes occurring within large bodies of water are not well understood. The radiative ¯uxes on both sides of the air±water interface must be measured in order to determine the net radiation at the surface. This is not easy to do in the ®eld. Therefore, the surface energy budget as expressed by Equation (2.1) may not be very useful or even appropriate to consider over water surfaces. A better alternative is to consider the energy budget of the whole energy-active water layer. 2.2.2 Energy budget of a layer An `ideal,' horizontally homogeneous, plane surface, which is also opaque to radiation, is rarely encountered in practice. More often, the earth's surface has horizontal inhomogeneities at small scale (e.g., plants, trees, houses, and

16

2 Energy Budget Near the Surface

building blocks), mesoscale (e.g., urban±rural di€erences, coastlines, hills, and valleys), and large scale (e.g., large mountain ranges). It may be partially transparent to radiation (e.g., water, tall grass, and crops). The surface may be sloping or undulating. In many practical situations, it will be more appropriate to consider the energy budget of a ®nite interfacial layer, which may include the small-scale surface inhomogeneities and/or the upper part of the subsurface medium which might be active in radiative exchanges. This layer must have ®nite mass and heat capacity which would allow the energy to be stored in or released from the layer over a given time interval. Such changes in the energy storage with time must, then, be considered in the energy budget for the layer. If the surface is relatively ¯at and homogeneous, so that the interfacial layer can be considered to be bounded by horizontal planes at the top and the bottom, one can still use a simpli®ed one-dimensional energy budget for the layer: RN = H + HL + HG + DHS

(2.4)

where DHS is the change in the energy storage per unit time, per unit horizontal area, over the whole depth of the layer. Thus, the main di€erence in the energy budget of an interfacial layer from that of an ideal surface is the presence of the rate of energy storage term DHS in the former. Strictly speaking, the various energy ¯uxes in Equation (2.4), except for DHS, are net vertical ¯uxes going in or out of the top and bottom faces of the layer. In reality, however, H and HL are associated only with the top surface (for an interfacial layer involving the water medium, the bottom boundary may be chosen so that there is no signi®cant radiative ¯ux in or out of the bottom surface), as shown schematically in Figure 2.2a. The rate of heat energy storage in the layer can be expressed as Z DHS ˆ

@ …rcT† dz @t

…2:5†

in which r is the mass density, c is the speci®c heat, T is the absolute temperature of the material at some level z, and the integral is over the whole depth of the layer. When the heat capacity of the medium can be assumed to be constant, independent of z, Equation (2.5) gives a direct relationship between the rate of energy storage and the rate of warming or cooling of the layer. The energy storage term DHS in Equation (2.4) may also be interpreted as the di€erence between the energy coming in (Hin) and the energy going out (Hout) of the layer, where Hin and Hout represent appropriate combinations of RN, H, HL, and HG, depending on their signs (Oke, 1987, Chapter 2). When the energy input to the layer exceeds the outgoing energy, there is a ¯ux convergence

2.2 Energy Balance Equations

17

Figure 2.2 Schematic representation of (a) energy budget of a layer, (b) ¯ux convergence, and (c) ¯ux divergence.

(DHS 4 0) which results in the warming of the layer. On the other hand, when energy going out exceeds that coming in, the layer cools as a result of ¯ux divergence (DHS 5 0). In the special circumstances of energy coming in exactly balancing the energy going out, there is no change in the energy storage of the layer (DHS = 0) or its temperature with time. These processes of vertical ¯ux convergence and divergence are schematically shown in Figure 2.2b and c. Example Problem 2 Over the tropical oceans the Bowen ratio is typically 0.1. Estimate the sensible and latent heat ¯uxes, as well as the rate of evaporation, in millimeters per day, from the ocean surface, when the net radiation received just above the surface is 400 W m72, the heat ¯ux to water below 50 m is negligible, the rate of warming of the 50-m-deep oceanic mixed layer is 0.058C day71, and the ocean surface temperature is 308C. Solution Here, RN = 400 W m72; B = H/HL = 0.1. From Equation (2.5), Z DHs ˆ rc

0

D

  @T @T dz ˆ rcD @t @z m

18

2 Energy Budget Near the Surface

where D = 50 m is the mixed layer depth, and (@T=@z†m = 0.058C day71 is the mean (layer-averaged) rate of warming of the mixed layer. Thus, DHS ˆ 1000 kg m

3

: 4:18  103 J kg

1

K

1

0:05 Ks 86 400

: 50 m :

1

ˆ 121 W m

2

Substituting in the energy budget Equation (2.4), one obtains H + HL = 259.1 W m72 With the Bowen ratio H/HL = 0.1, one obtains H = 25.4 W m72, and HL = 253.7 W m72. The rate of evaporation, E ˆ HL =Le ˆ

253:7 W m 2 2:45  106 J kg

ˆ 1:03  10

4

1

kg m

2

s

1

The evaporation rate in terms of the height of water column per unit time = E/rw = 8.9 mm day71. 2.2.3 Energy budget of a control volume In the energy budgets of an extensive horizontal surface and interfacial layer, only the vertical ¯uxes of energy are involved. When the surface under consideration is not ¯at and horizontal or when there are signi®cant changes (advections) of energy ¯uxes in the horizontal, then it would be more appropriate to consider the energy budget of a control volume. In principle, this is similar to the energy budget of a layer, but now one must consider the various energy ¯uxes integrated or averaged over the bounding surface of the control volume. The energy budget equation for a control volume can be expressed as ‡H L ‡ H  G ‡ DHS RN ˆ H

(2.6)

where the overbar over a ¯ux quantity denotes its average value over the entire area (A) of the bounding surface and the rate of storage is given by Z 1 @ …rcT† dV …2:7† DHS ˆ A @t in which the integration is over the control volume V.

2.3 Energy Budgets of Bare Surfaces

19

In practice, detailed measurements of energy ¯uxes are rarely available in order to evaluate their net contributions to the energy budget of a large irregular volume. Still, with certain simplifying assumptions, such energy budgets have been used in the context of several large ®eld experiments over the oceans, such as the 1969 Barbados Oceanographic and Meteorological Experiment (BOMEX) and the 1975 Air Mass Transformation Experiment (AMTEX). The concept of ¯ux convergence or divergence and its relation to warming or cooling of the medium is more general than depicted in Figure 2.2 for a horizontal layer. For a control volume, the direction of ¯ux is unimportant. According to Equation (2.6), the net convergence or divergence of energy ¯uxes in all directions determines the rate of energy storage and, hence, the rate of warming or cooling of the medium in the control volume (see Oke, 1987, Chapter 2). 2.3 Energy Budgets of Bare Surfaces A few cases of observed energy budgets will be discussed here for illustrative purposes only. It should be pointed out that relative magnitudes of the various terms in the energy budget may di€er considerably for other places, times, and weather conditions. Measured energy ¯uxes over a dry lake bed (desert) on a hot summer day are shown in Figure 2.3. This represents the simplest case of energy balance [Equation (2.1)] for a ¯at, dry, and bare surface in the absence of any evaporation or condensation (HL = 0). It is also an example of thermally extreme climatic environment; the observed maximum di€erence in the temperatures between the surface and the air at 2 m was about 288C.

Figure 2.3 Observed diurnal energy budget over a dry lake bed at El Mirage, California, on June 10±11, 1950. [After Vehrencamp (1953).]

20

2 Energy Budget Near the Surface

Over a bare, dry ground the net radiation is entirely balanced by direct heat exchanges with the surface by both the air and soil media. However, the relative magnitudes of these ¯uxes may change considerably from day to night, as indicated by measurements in Figure 2.3. Wetting of the subsurface soil by precipitation or arti®cial irrigation can dramatically alter the surface energy balance, as well as the microclimate near the surface. The daytime net radiation for the same latitude, season, and weather conditions becomes larger for the wet surface, because of the reduced albedo (re¯ectivity) and increased absorption of shortwave radiation by the surface. The latent heat ¯ux (HL) becomes an important or even dominant component of the surface energy balance [Equation (2.1)], while the sensible heat ¯ux to air (H) is considerably reduced. The latter may even become negative during early periods of irrigation, particularly for small areas, due to advective e€ects. This is called the `oasis e€ect' because it is similar to that of warm dry air blowing over a cool moist oasis. There is strong evaporation from the moist surface, resulting in cooling of the surface (due to latent heat transfer). The surface cooling causes a downward sensible heat ¯ux from the warm air to the cool ground. Thus, HL is positive, while H becomes negative, although much smaller in magnitude. The ground heat ¯ux may also be considerably reduced and even change sign if the surface is cooler than the soil below. Under such conditions, the latent heat ¯ux can be greater than the net radiation. The `oasis e€ect' of irrigation disappears as irrigation is applied over large areas and horizontal advections become less important. At later times, the relative importance of H and HL will depend on the soil moisture content and soil temperature at the surface. As the soil dries up, evaporation rate (E) and HL = LeE decrease, while the daytime surface temperature and the sensible heat ¯ux increase with time, for the same environmental conditions. Thus, the daytime Bowen ratio is expected to increase with time after precipitation or irrigation. A comparison of the surface energy budgets in a dry desert and an irrigated oasis has been given by Budyko (1958). The maximum daytime net radiation in the oasis is 40% larger than that in the dry desert and nearly all of the radiative input to the oasis goes into the latent heat of evaporation. Much smaller positive ground heat ¯ux is nearly balanced by the negative sensible heat ¯ux from air. 2.4 Energy Budgets of Canopies 2.4.1 Vegetation canopies The growth of vegetation over an otherwise ¯at surface introduces several complications into the energy balance. First, the ground surface is no longer the

2.4 Energy Budgets of Canopies

21

most appropriate datum for the surface energy balance, because the radiative, sensible, and latent heat ¯uxes are all spatially variable within the vegetative canopy. The energy budget of the whole canopy layer [Equation (2.4)] will be more appropriate to consider. For this, measurements of RN, H, and HL are needed at the top of the canopy (preferably, well above the tops of plants or trees where horizontal variations of ¯uxes may be neglected). Second, the rate of energy storage (DHS) consists of two parts, namely, the rate of physical heat storage and the rate of biochemical heat storage as a result of photosynthesis and carbon dioxide exchange. The latter may not be important on time scales of a few hours to a day, commonly used in micrometeorology. Nevertheless, the rate of heat storage by a vegetative canopy is not easy to measure or calculate. Third, the latent heat exchange occurs not only due to evaporation or condensation at the surface, but to a large extent due to transpiration from the plant leaves. The combination of evaporation and transpiration is called evapotranspiration; it produces nearly a constant ¯ux of water vapor above the canopy layer. An example of the observed energy budget over a barley ®eld on a summer day in England is shown in Figure 2.4. Note that the latent heat ¯ux due to evapotranspiration is the dominant energy component, which approximately balances the net radiation, while H and HG are an order of magnitude smaller. In the late afternoon and evening, HL even exceeded the net radiation and H became negative (downward heat ¯ux). The rate of heat storage was not measured, but is estimated (from energy balance) to be small in this case. Forest canopies have similar features to plant canopies, aside from the obvious di€erences in their sizes and stand architectures. Much larger heights of trees and the associated biomass of the forest canopy suggest that the rate of heat storage may not be insigni®cant, even over short periods of the order of a day.

Figure 2.4 Observed diurnal energy budget of a barley ®eld at Rothamsted, England, on July 23, 1963. [From Oke (1987); after Long et al. (1964).]

22

2 Energy Budget Near the Surface

Figure 2.5 Observed energy budget of a Douglas ®r canopy at Haney, British Columbia, on July 23, 1970. [From Oke (1987); after McNaughton and Black (1973).]

A typical example of the observed energy budget of a Douglas ®r canopy is given in Figure 2.5. In this case, DHS was roughly determined from the estimates of biomass and heat capacity of the trees and measurements of air temperatures within the canopy and, then, added to the measured ground heat ¯ux to get the combined HG + DHS term. This term is relatively small during the daytime, but of the same order of magnitude as the net radiation (RN) at night. Note that, during daytime, RN is more or less equally partitioned between the sensible and latent heat ¯uxes to the air. For other examples of measured energy budgets over forests, the reader may refer to Chapter 17 of Munn (1966). 2.4.2 Urban canopies Urban canopies are much more complex than vegetation canopies, because they represent the large diversity of size, shape, composition, and arrangement of the canopy or `roughness' elements including buildings, streets, trees, lawns and parks. A useful approach in describing an urban energy budget is to consider a near-surface active layer (urban canopy) or volume (box), whose top is set at or above roof level, and its base at the depth of zero net ground heat ¯ux over the chosen time scale or period (Oke, 1988). Then, one can neglect the extremely complex spatial arrangement of individual canopy elements as energy sources and sinks. The volume or box formulation restricts all energy ¯uxes, such as RN, H and HL, through its top. The internal heat storage change associated with all the canopy elements, the surrounding air and the ground, and internal heat sources due to fuel combustion can be represented as equivalent ¯uxes through unit area of the top of the box. Thus, an appropriate energy balance for the urban canopy can be expressed as (Oke, 1987, Chapter 8) RN + HF = H + HL + DHS

(2.8)

2.4 Energy Budgets of Canopies

23

in which HF is the equivalent heat ¯ux associated with fuel consumption within the urban area or its part (city center, suburb, etc.). Here, we have neglected the net advected heat ¯ow through the sides of the box, assuming that the volume under consideration is surrounded by similar land uses. In spite of the observed di€erences in turbidity, temperature, albedo, and emissivity between urban and rural surfaces, the net radiations are not found to be signi®cantly di€erent. But, the addition of the anthropogenic heat ¯ux HF to RN makes available substantially more total energy ¯ux for partitioning between the other ¯uxes on the right-hand side of Equation (2.8). The magnitude of HF in a city depends on its per capita energy use and its population density. Its relative importance in the energy budget of an urban canopy is indicated by the ratio HF/RN, which has been estimated for a number of cities in di€erent seasons (Oke, 1987, 1988). The largest values are found in densely inhabited cities in middle and high latitudes and the winter season. Annually averaged values of HF/RN for large cities vary between 0.2 (Los Angeles) and 3 (Moscow) with a more typical value of 0.35. Thus, the anthropogenic heat ¯ux is a signi®cant component of the energy budget of any large city. For a particular city, HF also displays large temporal (both seasonal and diurnal) and spatial variations. In all cases, the central business area or city center appears as the primary heat source, although there may be other localized `hot spots' in industrial areas. This suggests that HF can, perhaps, be neglected in the energy budgets of suburban residential areas. The importance of the sensible heat ¯ux to the energy budget of an urban canopy is well recognized. Built-up urban surfaces (e.g., roofs, streets, and highways) have low surface albedos (0.1±0.2) and readily absorb solar radiation during daytime. With their increased temperatures and enhanced turbulence activity in the urban canopy, a large portion of the available energy is transferred to the atmosphere as sensible heat. The impervious nature of these surfaces reduces the surface water availability for evaporation. Consequently, the latent heat ¯ux is relatively small and the Bowen ratio is large. But, locally, the situation is much di€erent over irrigated lawns or green parks. In urban areas, built-up impervious surfaces and natural pervious surfaces constitute two contrasting surface types with widely di€erent energy budgets on a local basis (Oke, 1987, Chapter 8). The importance of the heat storage term DHS in the urban canopy budget is also recognized. However, it is almost impossible to determine it from direct measurements, considering the many absorbing elements and surfaces involved in an urban canopy. In most experimental studies of energy budget, DHS is determined as the residual term from the energy balance equation. So far, such studies have been con®ned to suburban areas where measurements can be made with instruments placed on towers and roof tops.

24

2 Energy Budget Near the Surface

Figure 2.6 Monthly-averaged energy budgets at suburban and rural sites in Greater Vancouver, Canada, during summer. [From Oke (1987).]

An illustration of the suburban energy budget is shown in Figure 2.6 where it is compared with that of a nearby rural surface. The various ¯uxes shown are monthly averages for suburban and rural sites in Greater Vancouver, Canada, during summer (Oke, 1987, Chapter 8). These are based on direct measurements of RN and H, calculated DHS, and estimated HL as the residual in the energy balance equation. The suburban area is an area of fairly uniform one- or twostorey houses (36% built and 64% greenspace), while the `rural' site is an extensive area of grassland. For the latter, DHS essentially represents HG. As expected, the heat storage term over the suburban area is a signi®cant part of the energy budget, especially at nighttime. Some day-to-day variations of the energy budget of the same suburban area are shown by Oke (1988). Grimmond and Oke (1995) compare the summertime energy budgets of the suburbs for four North American cities (Chicago, Los Angeles, Sacramento, and Tucson) under clear, cloudy, and all sky conditions. As expected, the magnitudes of ¯uxes vary between cities; however, the diurnal trends of ¯ux partitioning are similar in terms of the timing of the peaks and changes in sign. Similar observations of the energy budgets of more heavily built city centers and commercial and industrial districts are lacking. However, we may anticipate both H and DHS playing a much larger role than in the suburbs, and HL playing a smaller role. Anthropogenic heat ¯ux HF is expected to be too large to

2.6 Applications

25

ignore, while the contribution of heat advection to or from surrounding areas may also have to be considered. 2.5 Energy Budgets of Water Surfaces Water covers more than two-thirds of the earth's surface. Therefore, it is important to understand the energy budget of water surfaces. It is complicated, however, by the fact that water is a ¯uid with a dynamically active surface and a surface boundary layer or mixed layer in which the motions are generally turbulent. Thus, convective and advective heat transfers within the surface boundary layer in water essentially determine HG. However, it is not easy to measure these transfers or HG directly. As discussed earlier, the radiation balance at the water surface is also complicated by the fact that shortwave radiation penetrates to considerable depth in water. Therefore, it will be more appropriate to consider the energy budget of a layer of water extending to a depth where both the convective and radiative heat exchanges become negligible. This will not be feasible, however, for shallow bodies of water with depths less than about 10 m. Even over large lakes and oceans, simultaneous measurements of all the energy ¯ux terms in Equation (2.4) on a short-term basis are lacking, largely because of the experimental diculties associated with ¯oating platforms and sea spray. The sensible and latent heat ¯uxes are the most frequently measured or estimated quantities. Over most ocean areas the latter dominates the former and the Bowen ratio is usually much less than unity (it may approach unity during periods of intense cold air advection over warm waters). The rate of heat storage in the oceanic mixed layer plays an important role in the energy budget. This layer acts as a heat sink (DHS 4 0) by day and a heat source (DHS 5 0) at night. The signi®cance of net radiation in the energy budget over water is not so clear. For some measurements of heat balance at sea on a daily basis in the course of a large experiment, the reader may refer to Kondo (1976). 2.6 Applications The following list includes some of the applications of energy balance at or near the earth's surface. . Prediction of surface temperature and frost conditions. . Indirect determination of the surface ¯uxes of heat (sensible and latent) to or from the atmosphere.

26

2 Energy Budget Near the Surface

. Estimation of the rate of evaporation from bare ground and water surfaces and evapotranspiration from vegetative surfaces. . Estimation of the rate of heat storage or loss by an oceanic mixed layer or a vegetative canopy. . Study of microclimates of the various surfaces. . Prediction of icing conditions on highways. In all these and possibly other applications, the various terms in the appropriate energy balance equation, except for the one to be estimated or predicted, have to be measured, calculated, or parameterized in terms of other measured quantities. Problems and Exercises 1. Describe the typical conditions in which you would expect the Bowen ratio to be as follows: (a) much less than unity; (b) much greater than unity; (c) negative. 2. (a) Over an ocean surface, the Bowen ratio is estimated to be about 0.2. Estimate the sensible and latent heat ¯uxes to the atmosphere, as well as the rate of evaporation, in millimeters per day, from the ocean surface, when the net radiation received just above the surface is 600 W m72, the heat ¯ux to the water below 50 m is negligible, the rate of warming of the 50 m deep oceanic mixed layer is 0.088C day71, and the sea surface temperature is 258C. (b) What will be the rate of warming or cooling of the 50 m deep oceanic mixed layer at the time of intense cold-air advection when the Bowen ratio is estimated to be 0.5, the net radiation loss from the surface is 50 W m72, and the rate of evaporation is 20 mm day71?

3. Giving schematic depictions, brie¯y discuss the energy balance of an extensive, uniform snowpack for the following conditions: (a) below-freezing air temperatures; (b) above-freezing air temperatures. 4. (a) Give a derivation of Equation (2.5) for the rate of heat storage in a soil layer. (b) Will the same expression apply to an oceanic mixed layer? Give reasons for your answer.

Problems and Exercises

27

5. Explain the following terms or concepts used in connection with the energy balance near the surface: (a) `ideal' surface; (b) evaporative cooling; (c) oasis e€ect; (d) ¯ux divergence. 6. (a) What are the major di€erences between the energy budgets of a bare soil surface and a vegetative surface? (b) How would you rewrite the energy balance of a subsurface soil layer and how does it di€er from that for the bare soil surface? 7. (a) Explain the processes and mechanisms involved in the increase of the surface temperature during morning hours after sun rise. (b) Is the surface energy balance equation (Equation 2.1) valid even when the surface temperature is rising rapidly with time? 8. (a) Explain the importance of the change in heat storage in the energy budget of an urban canopy. (b) What are the di€erent elements or components of DHS in Equation (2.8) and why is it so dicult to measure directly? 9. (a) How would you measure the rate of heat storage in the mixed layer of an ocean? (b) Discuss the mechanisms or processes involved in the daytime warming of the oceanic mixed layer and qualitatively compare it with that of a subsurface soil layer.

Chapter 3

Radiation Balance Near the Surface

3.1 Radiation Laws and De®nitions The transfer of energy by rapid oscillations of electromagnetic ®elds is called radiative transfer or simply radiation. These oscillations may be considered as traveling waves characterized by their wavelength or wave frequency cm/l, where cm is the wave speed in a given medium. All electromagnetic waves travel at the speed of light c % 3 6 108 m s71 in empty space and nearly the same speed in air (cm % c). There is an enormous range or spectrum of electromagnetic wavelengths or frequencies. Here we are primarily interested in the approximate range 0.1±100 mm, in which signi®cant contributions to the radiation balance of the atmosphere or the earth's surface occur. This represents only a tiny part of the entire electromagnetic wave spectrum. Of this, the visible light constitutes a very narrow range of wavelengths (0.40±0.76 mm). The radiant ¯ux density, or simply the radiative ¯ux, is de®ned as the amount of radiant energy (integrated over all wavelengths) received at or emitted by a unit area of the surface per unit time. The SI unit of radiative ¯ux is W m72, which is related to the CGS unit cal cm72 min71, commonly used earlier in meteorology as 1 cal cm72 min71 % 698 W m72. 3.1.1 Blackbody radiation laws Any body having a temperature above absolute zero emits radiation. If a body at a given temperature emits the maximum possible radiation per unit area of its surface, per unit time, at all wavelengths, it is called a perfect radiator or `blackbody.' The ¯ux of radiation (R) emitted by such a body is given by the Stefan±Boltzmann law R = sT 4

(3.1)

where s is the Stefan±Boltzmann constant = 5.67 6 1078 W m72 K74, and T is the surface temperature of the body in absolute (K) units. 28

3.1 Radiation Laws and De®nitions

29

Planck's law expresses the radiant energy per unit wavelength emitted by a blackbody as a function of its surface temperature Rl = (2phpc2/l5)[exp(hpc/blT) 71]71

(3.2)

where hp is Planck's constant = 6.626 6 10734 J s, and b is the Boltzmann constant = 1.381 6 10723 J K71. Note that the total radiative ¯ux is given by Z Rˆ

1 0

Rl d l

…3:3†

Equation (3.2) can be used to calculate and compare the spectra of blackbody radiation at various body surface temperatures. The wavelength at which Rl is maximum turns out to be inversely proportional to the absolute temperature and is given by Wien's law lmax = 2897/T

(3.4)

when lmax is expressed in micrometers. From the above laws, it is clear that the radiative ¯ux emitted by a blackbody varies in proportion to the fourth power of its surface temperature, while the wavelengths making the most contribution to the ¯ux, especially lmax, change inversely proportional to T. 3.1.2 Shortwave and longwave radiations The spectrum of solar radiation received at the top of the atmosphere is well approximated by the spectrum of a blackbody having a surface temperature of about 6000 K. Thus, the sun may be considered as a blackbody with an equivalent surface temperature of about 6000 K and lmax % 0.48 mm. The observed spectra of terrestrial radiation near the surface, particularly in the absence of absorbing substances such as water vapor and carbon dioxide, can also be approximated by the equivalent blackbody radiation spectra given by Equation (3.2). However, the equivalent blackbody temperature (Teb) is expected to be smaller than the actual surface temperature (T); the di€erence, T 7 Teb, depends on the radiative characteristics of the surface, which will be discussed later. Idealized or equivalent blackbody spectra of solar (Teb = 6000 K) and terrestrial (Teb = 287 K) radiations, both normalized with respect to their peak ¯ux per unit wavelength, are compared in Figure 3.1. Note that almost all the solar energy ¯ux is con®ned to the wavelength range 0.15±4.0 mm, while the

30

3 Radiation Balance Near the Surface

Figure 3.1 Calculated blackbody spectra of ¯ux density of solar and terrestrial radiation, both normalized by their peak ¯ux density. [From Rosenberg et al. (1983).]

terrestrial radiation is mostly con®ned to the range 3±100 mm. Thus, there is very little overlap between the two spectra, with their peak wavelengths (lmax) separated by a factor of almost 20. In meteorology, the above two ranges of wavelengths are characterized as shortwave and longwave radiations, respectively. 3.1.3 Radiative properties of natural surfaces Natural surfaces are not perfect radiators or blackbodies, but are, in general, gray bodies. They are generally characterized by several di€erent radiative properties, which are de®ned as follows. Emissivity is de®ned as the ratio of the energy ¯ux emitted by the surface at a given wavelength and temperature to that emitted by a blackbody at the same wavelength and temperature. In general, the emissivity may depend on the wavelength and will be denoted by el. For a blackbody, el = 1 for all wavelengths. Absorptivity is de®ned as the ratio of the amount of radiant energy absorbed by the surface material to the total amount of energy incident on the surface. In general, absorptivity is also dependent on wavelength and will be denoted by al. A perfect radiator is also a perfect absorber of radiation, so that al = 1 for a blackbody. Re¯ectivity is de®ned as the ratio of the amount of radiation re¯ected to the total amount incident upon the surface and will be denoted by rl. Transmissivity is de®ned as the ratio of the radiation transmitted to the subsurface medium to the total amount incident upon the surface and will be denoted by tl.

3.1 Radiation Laws and De®nitions

31

It is clear from the above de®nitions that al + r l + t l = 1

(3.5)

so that absorptivity, re¯ectivity, and transmissivity must have values between zero and unity. Kircho€'s law states that for a given wavelength, absorptivity of a material is equal to its emissivity, i.e., al = e l

(3.6)

Natural surfaces are in general radiatively `gray,' but in certain wavelength bands their emissivity or absorptivity may be close to unity. For example, in the so-called atmospheric window (8 to 14 mm wavelength band), water, wet soil, and vegetation have emissivities of 0.97±0.99. In studying the radiation balance for the energy budget near the earth's surface, we are more interested in the radiative ¯uxes integrated over all the wavelengths of interest, rather than in their wavelength-dependent spectral decomposition. For this, an overall or integrated emissivity and an integrated re¯ectivity, pertaining to radiation in a certain range of wavelengths, are used to characterize the surface. In particular, the term `albedo' is used to represent the integrated re¯ectivity of the surface for shortwave (0.15±4 mm) radiation, while the overall emissivity (e) of the surface refers primarily to the longwave (3± 100 mm) radiation. For natural surfaces, the emission of shortwave radiation is usually neglected, and the emission of longwave radiative ¯ux is given by the modi®ed Stefan±Boltzmann law RL =7esT 4

(3.7)

where the negative sign is introduced in accordance with our sign convention for radiative ¯uxes. Typical values of surface albedo (a) and emissivity (e) for di€erent types of surfaces are given in Table 3.1. The albedo of water is quite sensitive to the sun's zenith angle and may approach unity when the sun is near the horizon (rising or setting sun). It is also dependent on the wave height or sea state. To a lesser extent, the solar zenith angle is also found to a€ect albedos of soil, snow, and ice surfaces. Although snow has a high albedo, deposition of dust and aerosols, including man-made pollutants such as soot, can signi®cantly lower the albedo of snow. The albedo of a soil surface is strongly dependent on the wetness of the soil. Infrared emissivities of natural surfaces do not vary over a wide range; most surfaces have emissivities greater than 0.9, while only in a few cases values

32

3 Radiation Balance Near the Surface

Table 3.1 Radiative properties of natural surfaces. Surface type

Other speci®cations

Albedo (a)

Emissivity (e)

Water

Small zenith angle Large zenith angle Old Fresh Sea Glacier Dry Wet Dry clay Moist clay Wet fallow ®eld Concrete Black gravel road Long (1 m) Short (0.02 m) Wheat, rice, etc. Orchards Deciduous Coniferous

0.03±0.10 0.10±1.00 0.40±0.70 0.45±0.95 0.30±0.45 0.20±0.40 0.35±0.45 0.20±0.30 0.20±0.40 0.10±0.20 0.05±0.07 0.17±0.27 0.05±0.10 0.16 0.26 0.18±0.25 0.15±0.20 0.10±0.20 0.05±0.15

0.92±0.97 0.92±0.97 0.82±0.89 0.90±0.99 0.92±0.97

Snow Ice Bare sand Bare soil Paved Grass Agricultural Forests

0.84±0.90 0.91±0.95 0.95 0.97 0.71±0.88 0.88±0.95 0.90 0.95 0.90±0.99 0.90±0.95 0.97±0.98 0.97±0.99

Compiled from Sellers (1965), Kondratyev (1969), and Oke (1987).

become less than 0.8. Green, lush vegetation and forests are characterized by the highest emissivities, approaching close to unity. 3.2 Shortwave Radiation The ultimate source of all shortwave radiation received at or near the earth's surface is the sun. A large part of it comes directly from the sun, while other parts come in the forms of re¯ected radiation from the surface and clouds, and scattered radiation from atmospheric particulates or aerosols. 3.2.1 Solar radiation As a measure of the intensity of solar radiation, the solar constant (S0) is de®ned as the ¯ux of solar radiation falling on a surface normal to the solar beam at the outer edge of the atmosphere, when the earth is at its mean distance from the sun. An accurate determination of its value has been sought by many

3.2 Shortwave Radiation

33

investigators. The best estimate of S0 = 1368 W m72 is based on a series of measurements from high-altitude platforms such as the Solar Maximum Satellite; other values proposed in the literature fall in the range 1350± 1400 W m72. There are speculations that the solar constant may vary with changes in the sun spot activity, which has a predominant cycle of about 11 years. There are also suggestions of very long-term variability of the solar constant which may have been partially responsible for long-term climatic ¯uctuations in the past. The actual amount of solar radiation received at a horizontal surface per unit area over a speci®ed time is called insolation. It depends strongly on the solar zenith angle g and also on the ratio (d/dm) of the actual distance to the mean distance of the earth from the sun. The combination of the so-called inversesquare law and Lambert's cosine law gives the ¯ux density of solar radiation at the top of the atmosphere as R0 = S0 (dm/d)2 cos g

(3.8)

Then, insolation for a speci®ed period of time between t1 and t2 is given by Z I0 ˆ

t2

t1

R0 …t† dt

…3:9†

Thus, one can determine the daily insolation from Equation (3.9) by integrating the solar ¯ux density with time over the daylight hours. For a given calendar day/time and latitude, the solar zenith angle (g) and the ratio dm/d can be determined from standard astronomical formulas or tables, and the solar ¯ux density and insolation at the top of the atmosphere can be evaluated from Equations (3.8) and (3.9). These are also given in Smithsonian Meteorological Tables. The solar ¯ux density (Rs) and insolation (I) received at the surface of the earth may be considerably smaller than their values at the top of the atmosphere because of the depletion of solar radiation in passing through the atmosphere. The largest e€ect is that of clouds, especially low stratus clouds. In the presence of scattered moving clouds, Rs becomes highly variable. The second important factor responsible for the depletion of solar radiation is atmospheric turbidity, which refers to any condition of the atmosphere, excluding clouds, which reduces its transparency to shortwave radiation. The reduced transparency is primarily due to the presence of particulates, such as pollen, dust, smoke, and haze. Turbidity of the atmosphere in a given area may result from a combination of natural sources, such as wind erosion, forest ®res, volcanic eruptions, sea spray, etc., and various man-made sources of aerosols. Particles in the path of a solar beam re¯ect part of the radiation and scatter the other part.

34

3 Radiation Balance Near the Surface

Figure 3.2 Observed ¯ux density of solar radiation at the top of the atmosphere (curve A) and at sea level (curve B). The shaded areas represent absorption due to various gases in a clear atmosphere. [From Liou (1983.]

Large, solid particles re¯ect more than they scatter light, a€ecting all visible wavelengths equally. Therefore, the sky appears white in the presence of these particles. Even in an apparently clear atmosphere, air molecules and very small (submicrometer size) particles scatter the sun's rays. According to Rayleigh's scattering law, scattering varies inversely as the fourth power of the wavelength. Consequently, the sky appears blue because there is preferred scattering of blue light (lowest wavelengths of the visible spectrum) over other colors. Even in a cloud-free nonturbid atmosphere, atmospheric gases, such as oxygen, ozone, carbon dioxide, water vapor, and nitrous oxides, absorb solar radiation in selected wavelength bands. For example, much of the ultraviolet radiation is absorbed by stratospheric ozone and oxygen, and water vapor and carbon dioxide have a number of absorption bands at wavelengths larger than 0.8 mm. The combined e€ects of scattering and absorption of solar radiation in a clear atmosphere can be seen in Figure 3.2. Here, curve A represents the measured spectrum of solar radiation at the top of the atmosphere, curve B represents the measured spectrum at sea level, and the shaded areas represent the absorption of energy by various gases, primarily O3, O2, CO2, and H2O. The unshaded area between curve A and the outer envelope of the shaded area represents the depletion of solar radiation due to scattering. 3.2.2 Re¯ected radiation A signi®cant fraction (depending on the surface albedo) of the incoming shortwave radiation is re¯ected back by the surface. Shortwave re¯ectivities or

3.2 Shortwave Radiation

35

albedos for the various natural and man-made surfaces are given in Table 3.1. Knowledge of surface albedos and of their possible changes (seasonal, as well as long term) due to man's activities, such as deforestation, agriculture, and urbanization, has been of considerable interest to climatologists. Satellites now permit systematic and frequent observations of surface albedos over large regions of the world. Any inadvertent or intentional changes in the local, regional, or global albedo may cause signi®cant changes in the surface energy balance and hence in the micro- or macroclimate. For example, deliberate modi®cation of albedo by large-scale surface covering has been proposed as a means of increasing precipitation in certain arid regions. It has also been suggested that large, frequent oil spills in the ice- and snow-covered arctic region could lead to signi®cant changes in the regional energy balance and climate. Snow is the most e€ective re¯ector (high albedo) of shortwave radiation. On the other hand, water is probably the poorest re¯ector (low albedo), while ice falls between snow and water. Since large areas of earth are covered by water, sea ice, and snow, signi®cant changes in the snow and ice cover are likely to cause perceptible changes in the regional and, possibly, global albedo. Most bare rock, sand, and soil surfaces re¯ect 10±45% of the incident shortwave radiation, the highest value being for desert sands. Albedos of most vegetative surfaces fall in the range 10±25%. Wetting of the surface by irrigation or precipitation considerably lowers the albedo, while snowfall increases it. Albedos of vegetative surfaces, being sensitive to the solar elevation angle, also show some diurnal variations, with their minimum values around noon and maximum values near sunrise and sunset (see Rosenberg et al., 1983, Chapter 1). Shortwave re¯ectivities of clouds vary over a wide range, depending on the cloud type, height, and size, as well as on the angle of incidence of radiation (Welch et al., 1980). Thick stratus, stratocumulus, and nimbostratus clouds are good re¯ectors (a = 0.6±0.8, at normal incidence), large cumulus clouds are moderate re¯ectors (a = 0.2±0.5), and small cumuli are poor re¯ectors (a 5 0.2). Only a small part of solar radiation may reach the ground in cloudy conditions. A signi®cant part of the re¯ected radiation is likely to undergo multiple re¯ections between the surfaces and bases of clouds, thereby increasing the e€ective albedo of the surface. 3.2.3 Di€use radiation Di€use or sky radiation is that portion of the solar radiation that reaches the earth's surface after having been scattered by molecules and suspended particulates in the atmosphere. In cloudy conditions it also includes the portion

36

3 Radiation Balance Near the Surface

of the shortwave radiation which is re¯ected by the clouds. It is the incoming shortwave radiation in shade. Before sunrise and after sunset, all shortwave radiation is in di€use form. The ratio of di€use to the total incoming shortwave radiation varies diurnally, seasonally, and with latitude. In high latitudes, di€use radiation is very important; even in midlatitudes it constitutes 30±40% of the total incoming solar radiation. Cloudiness considerably increases the ratio of di€use to total solar radiation. 3.3 Longwave Radiation The longwave radiation RL received at or near the surface has two components: (1) outgoing radiation RL: from the surface, and (2) incoming radiation RL; from the atmosphere, including clouds. These are discussed separately in the following sections. 3.3.1 Terrestrial radiation All natural surfaces radiate energy, depending on their emissivities and surface temperatures, whose ¯ux density is given by Equation (3.7). As discussed earlier in Section 3.1, emissivities for most natural surfaces range from 0.9 to 1.0. Thus knowing the surface temperature and a crude estimate of emissivity, one can determine terrestrial radiation from Equation (3.7) to better than 10% accuracy. Diculties arise, however, in measuring or even de®ning the surface temperature, especially for vegetative surfaces. In such cases, it may be more appropriate to determine the apparent surface temperature, or equivalent blackbody temperature Teb from measurements of terrestrial radiation near the surface using Equation (3.1). In passing through the atmosphere, a large part of the terrestrial radiation is absorbed by atmospheric gases, such as water vapor, carbon dioxide, nitrogen oxides, methane, and ozone. In particular, water vapor and CO2 are primarily responsible for absorbing the terrestrial radiation and reducing its escape to the space (the so-called greenhouse e€ect). 3.3.2 Atmospheric radiation From our discussion earlier, it is clear that the atmosphere absorbs much of the longwave terrestrial radiation and a signi®cant part of the solar radiation. The atmospheric gases and aerosols which absorb energy also radiate energy,

3.3 Longwave Radiation

37

depending on the vertical distributions of their concentrations or mixing ratios and air temperature as functions of height. An important aspect of the atmospheric radiation is that absorption and emission of radiation by various gases occur in a series of discrete wavelengths or bands of wavelengths, rather than continuously across the spectrum, as shown in Figure 3.3. All layers of the atmosphere participate, to varying degrees, in absorption and emission of radiation, but the atmospheric boundary layer is most important in this, because the largest concentrations of water vapor, CO2, and other gases occur in this layer. Clouds, when present, are the major contributors to the incoming longwave radiation to the surface. They radiate like blackbodies (e % 1) at their respective cloud base temperatures. However, some of the radiation is absorbed by water vapor, CO2, and other absorbing gases before reaching the earth's surface. Computation of incoming longwave radiation from the atmosphere is tedious and complicated, even when the distributions of water vapor, CO2, cloudiness, and temperature are measured. It is preferable to measure RL; directly with an appropriate radiometer.

Figure 3.3 Absorption spectra of water vapor, carbon dioxide, oxygen and ozone, nitrogen oxide, methane and the atmosphere. [From Fleagle and Businger (1980).]

38

3 Radiation Balance Near the Surface

3.4 Radiation Balance Near the Surface The net radiation ¯ux RN in Equations (2.1) and (2.2) is a result of radiation balance between shortwave (RS) and longwave (RL) radiations at or near the surface, which can be written as RN = RS + RL

(3.10)

Further, expressing shortwave and longwave radiation balance terms as RS = RS; + RS:

(3.11)

RL = RL; + RL:

(3.12)

the overall radiation balance can also be written as RN = RS; + RS: + RL; + RL:

(3.13)

where the downward and upward arrows denote incoming and outgoing radiation components, respectively. The incoming shortwave radiation (RS;) consists of both the direct-beam solar radiation and the di€use radiation. It is also called insolation at the ground and can be easily measured by a solarimeter. It has strong diurnal variation (almost sinusoidal) in the absence of fog and clouds. The outgoing shortwave radiation (RS:) is actually the fraction of RS: that is re¯ected by the surface, i.e., RS: =7aRS;

(3.14)

where a is the surface albedo. Thus, for a given surface, the net shortwave radiation RS = (1 7 a)RS; is essentially determined by insolation at the ground. The incoming longwave radiation (RL;) from the atmosphere, in the absence of clouds, depends primarily on the distributions of temperature, water vapor, and carbon dioxide. It does not show a signi®cant diurnal variation. The outgoing terrestrial radiation (RL:), being proportional to the fourth power of the surface temperature in absolute units, shows stronger diurnal variation, with its maximum value in the early afternoon and minimum value at dawn. The two components are usually of the same order of magnitude, so that the net longwave radiation (RL) is generally a small quantity.

3.5 Observations of Radiation Balance

39

Under clear skies,|RL| 5 5 RS during the bright daylight hours and an approximate radiation balance is given by RN % RS = (1 7 a)RS;

(3.15)

which can be used to determine net radiation from simpler measurements or calculation of solar radiation at the surface. At nighttime, however, RS; = 0, and the radiation balance becomes RN = RL = RL; + RL:

(3.16)

Frequently at night RL; 57RL:, so that RN or RL is usually negative, implying radiative cooling of the surface. Near the sunrise and sunset times, all the components of the radiation balance are of the same order of magnitude and Equation (3.10) or (3.13) will be more appropriate than the simpli®ed Equation (3.15) or (3.16). Example Problem 1 The following radiation measurements were made over a dry, bare ®eld during a very calm and clear spring night: Outgoing longwave radiation from the surface = 400 W m72. Incoming longwave radiation from the atmosphere = 350 W m72. (a) Calculate the equivalent blackbody surface temperature, as well as the actual surface temperature if the surface emissivity is 0.95. (b) Estimate the ground heat ¯ux, making appropriate assumptions about other ¯uxes. Solution (a) RL: = sT 4eb =7400 W m72; s = 5.67 6 1078 W m72 K74, so that Teb = (400/s)1/4 % 289.8 K, is the equivalent blackbody temperature. Also, RL: =7esT 4S, so that TS = (400/es)1/4 % 293.6 K, is the actual surface temperature. (b) Using the energy budget equation (Equation 2.1), HG = RN 7 (H + HL). Very calm night implies H + HL % 0, so that HG = RN = RL: + RL;, using Equation (3.16). Thus, HG = 7 400 + 350 = 750 W m72, is the ground heat ¯ux. 3.5 Observations of Radiation Balance For the purpose of illustration, some examples of measured radiation balance over di€erent types of surfaces are provided. Figure 3.4 represents the various

40

3 Radiation Balance Near the Surface

Figure 3.4 Observed radiation budget over a 0.2 m stand of native grass at Matador, Saskatchewan, on 30 July 1971. [From Oke (1987); after Ripley and Redmann (1976).]

components of Equation (3.10) for a clear August day in England over a thick stand of grass, when the diurnal variation of grass temperature was about 208C. Note that in this case about one-fourth of the incident solar radiation was re¯ected back by the surface, while about three-fourths was absorbed. The net radiation is slightly less than RS, even during the midday hours, because of the net loss due to longwave radiation. The diurnal variation of RL; is much less than that of RL: (28% variation in RL: is consistent with the observed diurnal range of 208C in surface temperature). Figure 3.5 shows the diurnal variation of radiation budget components over Lake Ontario on a clear day in August. Note that the measured shortwave radiation (RS;) near the lake surface is only about two-thirds of the computed solar radiation (R0) at the top of the atmosphere. Of this, 25±30% is in the form of di€use-beam radiation (RD) at midday and the rest as direct-beam solar radiation (not plotted). The outgoing shortwave radiation (RS:) is relatively small, due to the low albedo (a % 0.07) of water. Both the incoming and outgoing longwave radiation components are relatively constant with time, due to small diurnal variations in the temperatures of the lake surface and the air above it. The net longwave radiation is a constant energy loss throughout the period of observations. The net radiation (RN) is dominated by RS; during the day and is equal to RL at night.

3.6 Radiative Flux Divergence

41

Figure 3.5 Observed radiation budget over Lake Ontario under clear skies on 28 August 1969. [From Oke (1987); after Davies et al. (1970).]

3.6 Radiative Flux Divergence The concept of energy ¯ux convergence or divergence and its relation to cooling or warming of a layer of the atmosphere of submedium has been explained in Chapter 1. Here, we discuss the signi®cance of net radiative ¯ux convergence or divergence to warming or cooling in the lowest layer of the atmosphere, namely, the PBL. The rate of warming or cooling of a layer of air due to change of net radiation with height can be calculated from the principle of conservation of energy. Considering a thin layer between the levels z and z + Dz, where the net radiative ¯uxes are RN(z) and RN (z + Dz), as shown in Figure 3.6, we get rcpDz(@T/@t)R = RN(z + Dz) 7 RN(z) = (@RN/@z)Dz or (@T/@t)R = (1/rcp)(@RN/@z)

(3.17)

where (@T/@t)R is the rate of change of temperature due to radiation and @RN/ @z represents the convergence or divergence of net radiation. Radiative ¯ux convergence occurs when RN increases with height (@RN/@z 4 0) and divergence occurs when @RN/@z 5 0. The former leads to warming and the latter to cooling of the air. Warming or cooling of air due to radiative ¯ux convergence or divergence is only a part of the total warming or cooling. Contributions of warm or cold air advection and turbulent exchange of sensible heat will be discussed later in Chapter 5. In the daytime, during clear skies, net radiation is dominated by the net shortwave radiation (RS), which usually does not vary with height in the PBL.

42

3 Radiation Balance Near the Surface

Figure 3.6 Schematic of radiative ¯ux convergence or divergence in the lower atmosphere.

Therefore, changes in air temperature with time due to radiation are insigni®cant or negligible; the observed daytime warming of the PBL is largely due to the convergence of sensible heat ¯ux. This may not be true, however, in the presence of fog and clouds in the PBL, when radiative ¯ux convergence or divergence may also become important. At night, the net radiation is entirely due to the net longwave radiation (RL). Both the terrestrial and atmospheric radiations and, hence, RL generally vary with height, because the concentrations of water vapor, CO2, and other gases that absorb and emit longwave radiation vary strongly with height in the PBL. Frequently, there is signi®cant radiative ¯ux divergence (which implies that radiative heat loss increases with height) within the PBL, especially during clear and calm nights. A signi®cant part of cooling in the PBL may be due to radiation, while the remainder is due to the divergence of sensible heat ¯ux. In many theoretical studies of the nocturnal stable boundary layer (SBL), radiative ¯ux divergence has erroneously been ignored; its importance in the determination of thermodynamic structure of the SBL has recently been pointed out (Garratt and Brost, 1981). In the presence of strong radiative ¯ux divergence or convergence, radiation measurements at some height above the surface may not be representative of their surface values. Corrections for the ¯ux divergence need to be applied to such measurements. Example Problem 2 The following measurements were made at night from a meteorological tower in the middle of a large farm: Net radiation at the 5 m level =775 W m72.

3.7 Applications

43

Net radiation at the 100 m level =7150 W m72. Sensible heat ¯ux at the surface =745 W m72. Planetary boundary layer height = 90 m. Calculate the average rate of cooling in the PBL due to the divergence of (a) net radiation and (b) sensible heat ¯ux. Solution (a) The rate of change of air temperature due to radiative ¯ux divergence 

@T @t

 R

ˆ

1 @RN 1 ˆ rcp @z 1200



150 ‡ 75 100 5

ˆ

6:58  10

ˆ

2:37 K h

4



Ks

1

1

(b) From the de®nition of the PBL, the sensible heat ¯ux at the top of the PBL (90 m) must be zero. The rate of change of air temperature due to sensible heat ¯ux divergence is given by   @T ˆ @t H

1 @H ˆ rcp @z

  1 0 ‡ 45 1200 90

ˆ

4:17  10

ˆ

1:50 K h

4

Ks

1

1

Thus, the total rate of cooling in the PBL was 3.87 K h71. 3.7 Applications The radiation balance at or near the earth's surface has the following applications: . Determining climate near the ground. . Determining radiative properties, such as albedo and emissivity of the surface. . Determining net radiation, which is an important component of the surface energy budget. . De®ning and determining the apparent (equivalent blackbody) surface temperature for a complex surface.

44

3 Radiation Balance Near the Surface

. Parameterizing the surface heat ¯uxes to soil and air in terms of net radiation. . Determining radiative cooling or warming of the PBL. These and other applications are particularly important in determining the microclimates of various natural and urban areas.

Problems and Exercises 1. The following measurements were made over a short grass surface on a winter night when no evaporation or condensation occurred: Outgoing longwave radiation from the surface = 365 W m72. Incoming longwave radiation from the atmosphere = 295 W m72. Ground heat ¯ux from the soil = 45 W m72. (a) Calculate the apparent (equivalent blackbody) temperature of the surface. (b) Calculate the actual surface temperature if surface emissivity is 0.92. (c) Estimate the sensible heat ¯ux to or from air.

2. (a) Estimate the combined sensible and latent heat ¯uxes from the surface to the atmosphere, given the following observations: Incoming shortwave radiation = 800 W m72. Heat ¯ux to the submedium = 150 W m72. Albedo of the surface = 0.35. (b) What would be the result if the surface albedo were to drop to 0.07 after irrigation? 3. The following measurements or estimates were made of the radiative ¯uxes over a short grass surface during a clear sunny day: Incoming shortwave radiation = 675 W m72. Incoming longwave radiation = 390 W m72. Ground surface temperature = 358C. Albedo of the surface = 0.20. Emissivity of the surface = 0.92. (a) From the radiation balance equation, calculate the net radiation at the surface. (b) What would be the net radiation after the surface is thoroughly watered so that its albedo drops to 0.10 and its e€ective surface temperature reduces to 258C? (c) Qualitatively discuss the e€ect of watering on other energy ¯uxes to or from the surface.

Problems and Exercises

45

4. Show that the variation of about 28% in terrestrial radiation in Figure 3.4 is consistent with the observed range of 10±308C in surface temperatures. 5. Explain the nature and causes of depletion of the solar radiation in passing through the atmosphere. 6. Discuss the consequences of the absorption of longwave radiation by atmospheric gases and the so-called greenhouse e€ect. 7. Discuss the merits of the proposition that net radiation RN can be deduced from measurements of solar radiation RS; during the daylight hours, using the empirical expression RN = ARS; + B where A and B are constants. On what factors are A and B expected to depend? 8. (a) Discuss the importance and consequences of the radiative ¯ux divergence at night above a grass surface. (b) If the net longwave radiation ¯uxes at 1 and 10 m above the surface are 7135 and 7150 W m72, respectively, calculate the rate of cooling or warming in 8C h71 due to radiation alone. 9. The following measurements were made at night from a meteorological tower: Net radiation at the 2 m level =7125 W m72. Net radiation at the 100 m level =7165 W m72. Sensible heat ¯ux at the surface =775 W m72. Planetary boundary layer height = 80 m. Calculate the average rate of cooling in the PBL due to the following: (a) the radiative ¯ux divergence; (b) the sensible heat ¯ux divergence.

Chapter 4

Soil Temperatures and Heat Transfer

4.1 Surface Temperature An `ideal' surface, de®ned in Chapter 2, may be considered to have a uniform surface skin temperature (Ts) which varies with time only in response to the time-dependent energy ¯uxes at the surface. An uneven or nonhomogeneous surface is likely to have spatially varying surface temperature. The surface temperature at a given location is essentially given by the surface energy balance, which in turn depends on the radiation balance, atmospheric exchange processes in the immediate vicinity of the surface, presence of vegetation or plant cover, and thermal properties of the subsurface medium. Here, we distinguish between the surface skin temperature and the nearsurface air temperature. The latter is measured at standard meteorological stations at the screen height of 1±2 m. The temperature at the air±soil interface will be referred to as the surface skin temperature or simply as the surface temperature (Ts). The direct in situ measurement of surface temperature is made very dicult by the extremely large temperature gradients that commonly occur near the surface in both the air (temperature gradients of 10±20 K mm71 in air are not uncommon very close to a bare heated surface) and the soil media, by the ®nite dimensions of the temperature sensor, and by the diculties of ventilating and shielding the sensor when it is placed at the surface. Therefore, the surface skin temperature is often determined by extrapolation of measured temperature pro®les in soil and air, with the knowledge of their expected theoretical behaviors. Another, perhaps better, method of determining the surface temperature, when emissivity of the surface is known, is through remote sensors, such as a downward-looking radiometer which measures the ¯ux of outgoing longwave radiation from the surface and, hence, Ts, using the modi®ed Stefan± Boltzmann Equation (3.7). If the surface emissivity is not known with sucient accuracy, this method will give the apparent (equivalent blackbody) surface temperature. In this way, surface or apparent surface temperatures are being routinely monitored by weather satellites. Measurements are made in narrow regions of the spectrum in which water vapor and CO2 are transparent. The 46

4.2 Subsurface Temperatures

47

technique has proved to be fairly reliable for measuring sea-surface temperatures, but not so reliable for land-surface temperatures. The times of minimum and maximum in surface temperatures as well as the diurnal range are of considerable interest to micrometeorologists. On clear days, the maximum surface temperature is attained typically an hour or two after the time of maximum insolation, while the minimum temperature is reached in early morning hours. The maximum diurnal range is achieved for a relatively dry and bare surface, under relatively calm air and clear skies. For example, on bare soil in summer, midday surface temperatures of 50±608C are common in arid regions, while early morning temperatures may be only 10± 208C. The bare-surface temperatures also depend on the texture of the soil. Fine-textured soils (e.g., clay) have greater heat capacities and smaller diurnal temperature range, as compared to coarse soils (e.g., sand). The presence of moisture at the surface and in the subsurface soil greatly moderates the diurnal range of surface temperatures. This is due to the increased evaporation from the surface, and also due to increased heat capacity and thermal conductivity of the soil. Over a free water surface, on the average, about 80% of the net radiation is utilized for evaporation. Over a wet, bare soil a substantial part of net radiation goes into evaporation in the beginning, but this fraction reduces as the soil surface dries up. Increased heat capacity of the soil further slows down the warming of the upper layer of the soil in response to radiative heating of the surface. The ground heat ¯ux is also reduced by evaporation. The presence of vegetation on the surface also reduces the diurnal range of surface temperatures. Part of the incoming solar radiation is intercepted by plant surfaces, reducing the amount reaching the surface. Therefore surface temperatures during the day are uniformly lower under vegetation than over a bare soil surface. At night the outgoing longwave radiation is also partly intercepted by vegetation, but the latter radiates energy back to the surface. This slightly slows down the radiative cooling of the surface. Vegetation also enhances the latent heat exchange due to evapotranspiration. It increases turbulence near the surface which provides more e€ective exchanges of sensible and latent heat between the surface and the overlying air. The combined e€ect of all these processes is to signi®cantly reduce the diurnal range of temperatures of vegetative surfaces. 4.2 Subsurface Temperatures Subsurface soil temperatures are easier to measure than the skin surface temperature. It has been observed that the range or amplitude of diurnal variation of soil temperatures decreases exponentially with depth and becomes

48

4 Soil Temperatures and Heat Transfer

Figure 4.1 Observed diurnal course of subsurface soil temperatures at various depths in a sandy loam soil with bare surface. [From Deacon (1969); after West (1952).]

insigni®cant at a depth of the order of 1 m or less. An illustration of this is given in Figure 4.1, which is based on measurements in a dry, sandy loam soil. Soil temperatures depend on a number of factors, which also determine the surface temperature. The most important are the location (latitude) and the time of the year (month or season), net radiation at the surface, soil texture and moisture content, ground cover, and surface weather conditions. Temperature may increase, decrease, or vary nonmonotonically with depth, depending on the season and the time of day. In addition to the `diurnal waves' present in soil temperatures in the top layer, daily or weekly averaged temperatures show a nearly sinusoidal `annual wave' which penetrates to much greater depths (*10 m) in the soil. This is illustrated in Figure 4.2. A simple theory for explaining the observed diurnal and annual temperature waves in soils and variations of their amplitudes and phase with depth will be presented later. First, we de®ne and discuss the thermal properties of soils which in¯uence soil temperatures and heat transfer. 4.3 Thermal Properties of Soils Thermal properties relevant to the transfer of heat through a medium and its e€ect on the average temperature or distribution of temperatures in the medium are the mass density, speci®c heat, heat capacity, thermal conductivity, and thermal di€usivity. The speci®c heat (c) of a material is de®ned as the amount of heat absorbed or released in raising or lowering the temperature of a unit mass of the material by

4.3 Thermal Properties of Soils

49

Figure 4.2 Annual temperature waves in the weekly averaged subsurface soil temperatures at two depths in a sandy loam soil. Fitted solid curves are sine waves. [From Deacon (1969); after West (1952).]

18. The product of mass density (r) and speci®c heat is called the heat capacity per unit volume, or simply heat capacity (C). Through solid media and still ¯uids, heat is transferred primarily through conduction, which involves molecular exchanges. The rate of heat transfer or heat ¯ux in a given direction is found to be proportional to the temperature gradient in that direction, i.e., the heat ¯ux in the z direction H =7k(@T/@z)

(4.1a)

in which the proportionality factor k is known as the thermal conductivity of the medium. The ratio of thermal conductivity to heat capacity is called the thermal di€usivity ah. Then, Equation (4.1) can also be written as H/rc =7 ah(@T/@z)

(4.1b)

where ah = k/rc = k/C. Heat is transferred through soil, rock, and other subsurface materials (except for water in motion) by conduction, so that the above-mentioned molecular thermal properties also characterize the submedium. Table 4.1 gives typical values of these for certain soils, as well as for the reference air and water media, for comparison purposes. Note that air has the lowest heat capacity, as well as the lowest thermal conductivity of all the natural materials, while water has the highest heat

Table 4.1 Molecular thermal properties of natural materialsa. Material

Condition

Mass density r (kg m73 6 103)

Speci®c heat c (J kg71 K71 6 103)

Heat capacity C (J m73 K71 6 106)

Thermal conductivity k (W m71 K71)

Thermal di€usivity ah (m2 s71 6 1076

Air Water Ice Snow Snow Sandy soil (40% pore space) Clay soil (40% pore space) Peat soil (80% pore space) Rock

208C, Still 208C, Still 08C, Pure Fresh Old Fresh Saturated Dry Saturated Dry Saturated Solid

0.0012 1.00 0.92 0.10 0.48 1.60 2.00 1.60 2.00 0.30 1.10 2.70

1.01 4.18 2.10 2.09 2.09 0.80 1.48 0.89 1.55 1.92 3.65 0.75

0.0012 4.18 1.93 0.21 0.84 1.28 2.96 1.42 3.10 0.58 4.02 2.02

0.025 0.57 2.24 0.08 0.42 0.30 2.20 0.25 1.58 0.06 0.50 2.90

20.5 0.14 1.16 0.38 0.05 0.24 0.74 0.18 0.51 0.10 0.12 1.43

a

After Oke (1987) and Garratt (1992).

4.4 Theory of Soil Heat Transfer

51

capacity. Thermal di€usivity of air is very large because of its low density. Thermal properties of both air and water depend on temperature. Most soils consist of particles of di€erent sizes and materials with a signi®cant fraction of pore space, which may be ®lled with air and/or water. Thermal properties of soils, therefore, depend on the properties of the solid particles and their size distribution, porosity of the soil, and the soil moisture content. Of these, the soil moisture content is a short-term variable, changes of which may cause signi®cant changes in heat capacity, conductivity, and di€usivity of soils (Oke, 1987, Chapter 2; Rosenberg et al., 1983, Chapter 2). Addition of water to an initially dry soil increases its heat capacity and conductivity markedly, because it replaces air (a poor heat conductor) in the pore space. Both the heat capacity and thermal conductivity are monotonic increasing functions of soil moisture content. The former increases almost linearly with the soil moisture content, while the latter approaches a constant value with increasing soil moisture. Their ratio, thermal di€usivity, for most soils increases with moisture content initially, attains a maximum value, and then falls o€ with further increase in soil moisture content. It will be seen later that thermal di€usivity is a more appropriate measure of how rapidly surface temperature changes are transmitted to deeper layers of submedium. Soils with high thermal di€usivities allow rapid penetration of surface temperature changes to large depths. Thus, for the same diurnal heat input or output, their temperature ranges are less than for soils with low thermal di€usivities.

4.4 Theory of Soil Heat Transfer Here we consider a uniform conducting medium (soil), with heat ¯owing only in the vertical direction. Let us consider the energy budget of an elemental volume consisting of a cylinder of horizontal cross-section area DA and depth Dz, bounded between the levels z and z + Dz, as shown in Figure 4.3. Heat ¯ow in the volume at depth z = HDA. Heat ¯ow out of the volume at z + Dz = [H + (@H/@z) Dz]DA. The net rate of heat ¯ow in the control volume =7(@H/@z)Dz DA. The rate of change of internal energy within the control volume = (@/@t) (DADzCsT ), where Cs is the volumetric heat capacity of soil. According to the law of conservation of energy, if there are no sources or sinks of energy within the elemental volume, the net rate of heat ¯owing in the volume should equal the rate of change of internal energy in the volume, so that (@/@t)(CsT ) =7@H/@z

(4.2)

52

4 Soil Temperatures and Heat Transfer

Figure 4.3 Schematic of heat transfer in a vertical column of soil below a ¯at, horizontal surface.

Further, assuming that the heat capacity of the medium does not vary with time, and substituting from Equation (4.1) into (4.2), we obtain Fourier's equation of heat conduction: @T/@t = (@/@z) [(k/Cs)(@T/@z)] = (@/@z)[ah(@T/@z)]

(4.3)

The one-dimensional heat conduction equation derived here can easily be generalized to three dimensions by considering the net rate of heat ¯ow in an elementary control volume DxDyDz from all the directions. In our applications involving heat transfer through soils, however, we will be primarily concerned with the one-dimensional Equation (4.2) or (4.3). Equation (4.2) can be used to determine the ground heat ¯ux HG in the energy balance equation from measurements of soil temperatures as functions of time, at various depths below the surface. The method is based on the integration of Equation (4.2) from z = 0 to D Z HG ˆ HD ‡

0

D

@ …Cs T† dz @t

…4:4†

where D is some reference depth where the soil heat ¯ux HD is either zero (e.g., if at z = D, @T/@z = 0) or can be easily estimated [e.g., using Equation (4.1)]. The former is preferable whenever feasible, because it does not require a knowledge of thermal conductivity, which is more dicult to measure than heat capacity. 4.5 Thermal Wave Propagation in Soils The solution of Equation (4.3), with given initial and boundary conditions, is used to study theoretically the propagation of thermal waves in soils and other

4.5 Thermal Wave Propagation in Soils

53

substrata. For any arbitrary prescription of surface temperature as a function of time and soil layers, the solution to Equation (4.3) can be obtained numerically. Much about the physics of thermal wave propagation can be learned, however, from a simple analytic solution which is obtained when the surface temperature is speci®ed as a sinusoidal function of time and the subsurface medium is assumed to be homogeneous throughout the depth of wave propagation Ts = Tm + As sin[(2p/P)(t 7 tm)]

(4.5)

Here, Tm is the mean temperature of the surface or submedium, As and P are the amplitude and period of the surface temperature wave, and tm is the time when Ts = Tm, as the surface temperature is rising. The solution of Equation (4.3) satisfying the boundary conditions that at z = 0, T = Ts(t), and as z ? ?, T ? Tm, is given by T = Tm + As exp(7z/d ) sin[(2p/P)(t 7 tm) 7 z/d ]

(4.6)

which the reader may verify by substituting in Equation (4.3). Here, d is the damping depth of the thermal wave, de®ned as d = (Pah/p)1/2

(4.7)

Note that the period of thermal wave in the soil remains unchanged, while its amplitude decreases exponentially with depth (A = As exp(7z/d )); at z = d the wave amplitude is reduced to about 37% of its value at the surface and at z = 3d the amplitude decreases to about 5% of the surface value. The phase lag relative to the surface wave increases in proportion to depth (phase lag = z/d ), so that there is a complete reversal of the wave phase at z = pd. The corresponding lag in the time of maximum or minimum in temperature is also proportional to depth (time lag = zP/2pd ). The results of the above simple theory would be applicable to the propagation of both the diurnal and annual temperature waves through a homogeneous submedium, provided that the thermal di€usivity of the medium remains constant over the whole period, and the surface temperature wave is nearly sinusoidal. For the diurnal period, the latter condition is usually not satis®ed especially during the nighttime period when the wave is observed to be more asymmetric around its minimum value (Rosenberg et al., 1983, Chapter 2). Also, disturbed weather conditions with clouds and precipitation are likely to alter the surface temperature wave, as well as thermal di€usivity due to changes in the soil moisture content. For the annual period, the assumption of constant thermal di€usivity may be even more questionable, except for bare soils in arid regions. Note that the damping depth, which is a measure of the extent of

54

4 Soil Temperatures and Heat Transfer

thermal wave propagation, for the annual wave is expected to be times the damping depth for the diurnal wave.

p 365 = 19.1

Example Problem 1 Calculate the damping depth and the depth of thermally active soil layer where there is a complete reversal of the phase of the diurnal thermal wave from that at the surface for the following types of soils: (a) dry sandy soil; (b) saturated sandy soil (40% pore space); (c) dry clay soil; (d) dry peat soil. What will be the amplitude of the thermal wave at the depth of the phase reversal relative to that at the surface? Solution Using the thermal di€usivities given in Table 4.1, Equation (4.7) for damping depth d, and the depth of phase reversal as pd, one can obtain the following results: (a) For dry sandy soil, ah = 0.24 6 1076 m2 s71. Damping depth d = (24 6 3600 6 0.24 6 1076/p)1/2 = 0.081 m. Depth of phase reversal = pd = 0.25 m. (b) For saturated sandy soil, ah = 0.74 6 1076 m2 s71. Damping depth d = 0.143 m. Depth of phase reversal = pd = 0.45 m. (c) For dry clay soil, ah = 0.18 6 1076 m2 s71. Damping depth d = 0.070 m. Depth of phase reversal = pd = 0.22 m. (d) For dry peat soil, ah = 0.10 6 1076 m2 s71. Damping depth d = 0.052 m. Depth of phase reversal = pd = 0.16 m. In all the cases the amplitude of the thermal wave at z = pd, relative to that at the surface, is A/As = exp(7p d/d ) = exp(7p) % 0.042 One can conclude that dry peat soil o€ers the maximum resistance (minimum d) and saturated sandy soil the minimum resistance (maximum d ) to the propagation of the diurnal thermal wave. The observed temperature waves (Figures 4.1 and 4.2) in bare, dry soils are found to conform well to the pattern predicted by the theory. In particular, the observed annual waves show nearly perfect sinusoidal forms. The plots of wave amplitude (on log scale) and phase or time lag as functions of depth (both on linear scale) are well represented by straight lines (Figure 4.4) whose slopes determine the damping depth and, hence, thermal di€usivity of the soil.

4.5 Thermal Wave Propagation in Soils

55

Figure 4.4 Variations of amplitude and time lag of the annual soil temperature waves with depth in the soil. [From Deacon (1969).]

Example Problem 2 Using the weekly averaged data on soil temperatures at di€erent depths obtained by West (1952), the amplitude and time lag of thermal waves are plotted as functions of depth in soil in Figure 4.4. Estimate the damping depth and thermal di€usivity of the soil from the best-®tted lines through the data points. Solution Note that according to Equation (4.6), the amplitude of thermal wave decreases exponentially with depth, i.e., A ˆ As exp… z=d† or

ln A ˆ ln As

z d

Therefore, a plot of ln A (A on a log scale) against z should result in a straight line with a slope of 71/d. The slope of the best-®tted straight line through the amplitude data in Figure 4.4 can be estimated, from which d % 2.05 m. Thermal di€usivity ah = pd2/P % 0.42 6 1076 m2 s71. Also, according to the thermal wave equation (Equation 4.6), Time lag = Pz/2pd

56

4 Soil Temperatures and Heat Transfer

Therefore, a plot of time lag versus z should also result in a straight line with a slope of P/2pd. The slope of the best-®tted line throughout the time lag data points in Figure 4.4 can be estimated as 25.6 days m71, from which d = P/(2p 6slope) % 2.27 m ah = pd 2/P % 0.51 6 1076 m2 s71 The two estimates of d, based on amplitude and time lag data, are in fairly good agreement. The agreement between the estimated damping depths based on the top 0.30 m of the soil temperature data for the diurnal period is found to be much better (Deacon, 1969). 4.6 Measurement and Parameterization of Ground Heat Flux 4.6.1 Measurement by heat ¯ux plate There is no direct method of measuring the ground heat ¯ux HG (the heat ¯ux through soil at the surface). Theoretically, the heat conduction equation [Equation (4.1a)] is valid right up to the surface. In practice, however, the diculty of accurately measuring the near-surface soil temperature gradient and soil thermal conductivity renders this approach impractical. Even at a small depth below the surface, where the soil temperature gradient can be accurately measured, thermal conductivity of the soil is highly variable and not easy to measure. However, heat ¯ux plates are often used to measure the soil heat ¯ux directly. The plate consists of material of known thermal conductivity, and temperature di€erence across its lower and upper faces is measured by a di€erential thermopile. The faces are often covered by thin metal plates in order to protect them and to ensure good thermal contact with the soil. The heat ¯ux plate should be small, so that it does not disrupt the soil and has a low heat capacity for quick response to any changes with time. In a ®eld experiment, several heat ¯ux plates may be buried horizontally in the soil at a depth of 10± 50 mm. Thus, the average soil heat ¯ux H1 at a small depth z1 can be measured. In order to estimate HG from the measurement of H1, one can use Equation (4.4) with D = z1 so that Z HG ˆ H1 ‡

z1 0

@ …Cs T†dz ˆ H1 ‡ DHs @t

…4:8†

where DHs is the rate of heat storage in the top soil layer of depth z1. For the thin soil layer, the above heat storage term will be signi®cant only when there is rapid

4.6 Measurement and Parameterization of Ground Heat Flux

57

warming or cooling of the top soil layer (e.g., around the times of sunrise and sunset). The average warming or cooling rate can be determined from the continuous measurement of soil temperature as a function of time in the middle of the layer. An indirect method of estimating HG from the measurement of soil temperatures at various depths, at regular time intervals, has already been discussed in Section 4.4. It is based on the application of Equation (4.4) in which the last term represents the rate of heat storage in a deeper layer of depth D. The soil heat ¯ux at the bottom of this layer is also estimated, rather than measured with a heat ¯ux plate.

4.6.2 Simple parameterizations In some climate and general circulation models, the ground heat ¯ux is simply neglected. The rationale for this is provided by the observation that, averaged over the entire diurnal cycle, the net ground heat ¯ux is often near zero. This is because the heating of the thermally active soil layer during the day is approximately balanced by the cooling at night. For shorter periods used in micrometeorology, however, HG is a signi®cant component of the energy balance near the surface, especially at night. A simple parameterization of the same, discussed in Chapter 2, is to specify HG as a constant fraction of the net radiation RN during the day and a di€erent (larger) fraction at night. The actual values of the ratio HG/RN for the day and night periods depend on the surface and soil characteristics and are determined empirically. Alternatively, the ground heat ¯ux can also be assumed as a constant fraction of the sensible heat ¯ux into air. Both of these simple parameterizations imply that the sign of the ground heat ¯ux is always the same as that of net radiation or sensible heat ¯ux in air. This is not particularly true during the morning and evening transition periods when di€erent energy ¯uxes change signs at di€erent times. These simple parameterizations are also not applicable over water surfaces.

4.6.3 Soil heat transfer models An expression for the ground heat ¯ux can be obtained from Equation (4.6) as HG ˆ

   @T 2p 1=2 …t k ˆ …2pCs k=P† As sin @z zˆ0 P

p tm † ‡ 4

 …4:9†

58

4 Soil Temperatures and Heat Transfer

which predicts that the maximum in surface temperature should lag behind the maximum in ground heat ¯ux by P/8, or 3 h for the diurnal period and 1.5 months for the annual period. The above equation also indicates that the amplitude of ground heat ¯ux wave is proportional to the square root of the product of heat capacity and thermal conductivity and inversely proportional to the square root of the period; it is also proportional to the amplitude of the surface temperature wave. The above predicted sinusoidal variation of the ground heat ¯ux is not usually consistent with direct measurements or indirect estimates of HG from the energy budget equation. In particular, the nighttime variation of HG with time is much ¯atter or more nearly uniform (see Munn, 1966, Chapter 5). There are several reasons for this disagreement. First, the assumed sinusoidal variation of the surface temperature in Equation (4.5), which constitutes the upper boundary condition for the solution of Equation (4.3), may not be realistic. Secondly, the simplifying assumption of the homogeneity of the submedium is often not realized. Thermal properties may vary with depth because of the layer structure of the submedium, variations in the soil moisture content, and presence of roots and surface vegetation. Thermal properties may also vary with time in response to irrigation, precipitation, and evaporation. In such cases, numerical models of soil heat and moisture transfer with varying degrees of complication are used, in conjunction with the surface energy balance equation. 4.6.4 Force±restore method A more widely used parameterization of the energy exchange with the subsurface medium in atmospheric models is based on the two-layer approximation of the submedium, in which a shallow thermally active layer of soil overlies a thick constant-temperature slab. The depth dg of the near-ground layer is carefully chosen, based on knowledge of the damping depth. The lower layer is assumed to be of sucient thickness that, over the time scales of interest, the soil heat ¯ux at the bottom is zero. The soil heat ¯ux at z = dg can be parameterized as Hg = mg(Tg 7 Tm)

(4.10)

where Tg and Tm are the layer-averaged temperatures of the upper and lower layers, respectively, and mg is a heat transfer coecient. The energy balance equation near the surface, including the near-ground soil layer of depth dg, can be written as RN = H + HL + Hg + DHs

(4.11)

4.6 Measurement and Parameterization of Ground Heat Flux

59

in which the rate of energy storage can be expressed as DHs ˆ Cs dg

@Tg @Tg ˆ Cg @t @t

…4:12†

where Cg = Csdg is the heat capacity per unit area of the near-ground soil layer. Substituting from Equations (4.10) and (4.12) into (4.11), one obtains a prognostic equation for the temperature of the near-ground layer as Cg

@Tg ˆ …RN @t

H

HL †

mg …Tg

Tm †

…4:13†

This prognostic equation for the ground temperature is generally referred to as the force±restore method, because the net radiative and atmospheric forcing (RN 7 H 7 HL) at the surface is modi®ed by the deep soil ¯ux which tends to restore the near-ground temperature to that of deep soil. Note that, if the surface forcing term is removed, the restoring term in Equation (4.13) will cause Tg to move exponentially towards Tm. Equation (4.13) has simply been extended to predict the diurnal variation of the ground surface temperature as Cg

@Ts ˆ …RN @t

H

HL † ˆ ms …Ts

Tm †

…4:14†

in which the soil properties are appropriately chosen or speci®ed as (Garratt, 1992, Chapter 8) Cg = Cs(ah/2O)1/2

(4.15)

ms = OCg

(4.16)

where O is the angular rotational speed of the earth. Note that Equation (4.15) with Cg = Csdg implies that dg = d/2. Thus, the most appropriate depth of the near-ground layer in the above two-layer model is half of the damping depth for the diurnal period P = 2p/O. The above expressions for Cg and ms are strictly based on the assumption of a sinusoidal surface temperature and, hence, ground heat ¯ux forcing, as expressed in Equations (4.5) and (4.9), respectively. Since the surface forcing is not really sinusoidal in nature, slightly modi®ed expressions Cg = 0.95Cs(ah/2O)1/2

(4.17)

ms = 1.18OCg

(4.18)

60

4 Soil Temperatures and Heat Transfer

are more commonly used in numerical prediction models of the ground surface temperature (Garratt, 1992).

4.7 Applications Some of the applications of the knowledge of surface and soil temperatures and soil heat transfer are as follows: . . . . . .

Study of surface energy budget and radiation balance. Prediction of surface temperature and frost conditions. Determination of the rate of heat storage or release by the submedium. Study of microenvironment of plant cover, including the root zone. Environmental design of underground structures. Determination of the depth of permafrost zone in high latitudes.

Problems and Exercises 1. You want to measure the temperature of a bare soil surface using ®ne temperature sensors on a summer day when the sensible heat ¯ux to air might be typically 600 W m72 and the ground heat ¯ux 100 W m72. Will you do this by extrapolation of measurements near the surface in air or in soil? Give reasons for your choice. Also estimate temperature gradients near the surface in both the air (k = 0.03 W m71 K71) and the soil (k = 0.30 W m71 K71). 2. Discuss the e€ect of vegetation or plant cover on the diurnal range of surface temperatures, as compared to that over a bare soil surface, and explain why green lawns are much cooler than roads and driveways on sunny afternoons. 3. What is the physical signi®cance of the damping depth in the propagation of thermal waves in a submedium and on what factors does it depend? 4. What are the basic assumptions underlying the simple theory of soil heat transfer presented in this chapter? Discuss the situations in which these assumptions may not be valid. 5. By substituting from Equation (4.6) into (4.3), verify that the former is a solution of Fourier's equation of heat conduction through a homogeneous soil medium, and then, derive the expression [Equation (4.9)] for the ground heat ¯ux.

Problems and Exercises

61

6. Compare and contrast the alternative methods of determining the ground heat ¯ux from measurements of soil temperatures and thermal properties. 7. The following soil temperatures (8C) were measured during the Great Plains Field Program at O'Neil, Nebraska: Depth (m) Time (h) 0435 0635 0835 1035 1235 1435 1635 1835 2035 2235 0035 0235 0435

Day August 31

September 1

0.025

0.05

0.10

0.20

0.40

25.54 24.84 25.77 29.42 33.25 35.25 34.84 32.63 30.07 28.42 27.09 26.09 25.30

26.00 25.30 25.35 27.36 30.32 32.63 33.20 32.05 30.20 28.74 27.50 26.60 25.83

26.42 25.84 25.42 25.98 27.62 29.52 30.62 30.62 29.91 28.84 27.84 27.06 26.40

26.32 25.97 25.56 25.39 25.57 26.11 26.88 27.41 27.68 27.57 27.22 26.87 26.53

24.50 24.48 24.36 24.34 24.27 24.24 24.26 24.32 24.47 24.64 24.73 24.78 24.84

(a) Plot on a graph temperature waves as functions of time and depth. (b) Plot on a graph the vertical soil temperature pro®les at 0435, 0835, 1235, 1635, 2035, and 0035 h. (c) Determine the damping depth and thermal di€usivity of the soil from the observed amplitudes, as well as from the times of temperature maxima (taken from the smoothed temperature waves) as functions of depth. (d) Estimate the amplitude of the surface temperature wave and the time of maximum surface temperature from extrapolation of the soil temperature data. 8. From the soil temperature data given in the above problem, calculate the ground heat ¯ux at 0435 and 1635 h, using Equation (4.4) with the measured soil heat capacity of 1.33 6 106 J m73 K71 and the average value of thermal di€usivity determined in Problem 7(c) above.

Chapter 5

Air Temperature and Humidity in the PBL

5.1 Factors In¯uencing Air Temperature and Humidity The following factors and processes in¯uence the vertical distribution (pro®le) of air temperature in the atmospheric boundary layer. . Type of air mass and its temperature just above the PBL, which depend on the synoptic situation and the large-scale circulation pattern. . Thermal characteristics of the surface and submedium, which in¯uence the diurnal range of surface temperatures. . Net radiation at the surface and its variation with height, which determine the radiative warming or cooling of the surface and the PBL. . Sensible heat ¯ux at the surface and its variation with height, which determine the rate of warming or cooling of the air due to convergence or divergence of sensible heat ¯ux. . Latent heat exchanges during evaporation and condensation processes at the surface and in air, which in¯uence the surface and air temperatures, respectively. . Warm or cold air advection as a function of height in the PBL. . Height or the PBL to which turbulent exchanges of heat are con®ned. Similarly, the factors in¯uencing the speci®c humidity or mixing ratio of water vapor in the PBL are as follows: . Speci®c humidity of air mass just above the PBL. . Type of surface, its temperature, and availability of moisture for evaporation and/or transpiration. . The rate of evapotranspiration or condensation at the surface and the variation of water vapor ¯ux with height in the PBL. . Horizontal advection of water vapor as a function of height. . Mean vertical motion in the PBL and possible cloud formation and precipitation processes. . The PBL depth through which water vapor is mixed. 62

5.2 Basic Thermodynamic Relations and the Energy Equation

63

The vertical exchanges of heat and water vapor are primarily through turbulent motions in the PBL; the molecular exchanges are important only in a very thin (of the order of 1 mm or less) layer adjacent to the surface. In the atmospheric PBL, turbulence is generated mechanically (due to surface friction and wind shear) and/or convectively (due to surface heating and buoyancy). These two types of turbulent motions are called forced convection and free convection, respectively. The combination of the two is sometimes referred to as mixed convection, although more often the designation `forced' or `free' is used, depending on whether a mechanical or convective exchange process dominates. The convergence or divergence of sensible heat ¯ux leads to warming or cooling of air, similar to that due to net radiative ¯ux divergence or convergence. The rate of warming is equal to the vertical gradient of 7H/rcp, i.e., @T/@t = 7(@H/@z)/rcp. Thus a gradient of 1 W m73 in sensible heat ¯ux will produce a change in air temperature at the rate of about 38C h71. Similarly, divergence or convergence of water vapor ¯ux leads to decrease or increase of speci®c humidity with time. Horizontal advections of heat and moisture become important only when there are sharp changes in the surface characteristics (e.g., during the passage of a front) in the horizontal. Mean vertical motions forced by changes in topography often lead to rapid changes in temperature and humidity of air, as well as to local cloud formation and precipitation processes, as moist air rises over the mountain slopes. Equally sharp changes occur on the lee sides of mountains. Even over a relatively ¯at terrain, the subsidence motion often leads to considerable warming and drying of the air in the PBL. 5.2 Basic Thermodynamic Relations and the Energy Equation The pressure in the lower atmosphere decreases with height in accordance with the hydrostatic equation @P/@z =7rg

(5.1)

Equation (5.1) is strictly valid only for the atmosphere at rest. It is also a good approximation for large-scale atmospheric ¯ows. For small mesoscale and microscale motions, however, the deviations from the hydrostatic equation due to vertical motions and accelerations may also become signi®cant. But, in thermodynamical relations discussed here, the atmosphere is assumed to be in a hydrostatic equilibrium. Another fundamental relationship between the commonly used thermodynamic variables is the equation of state or the ideal gas law P = RrT = (R /m)rT

(5.2)

64

5 Air Temperature and Humidity in the PBL

in which R is the speci®c gas constant, R = 8.314 J K71 mole71 is the absolute gas constant, and m is the mean molecular mass of air. Some of the thermodynamic relations follow from the ®rst law of thermodynamics which is essentially a statement about the conservation of energy. When applied to an air parcel, the law states that an increase in the internal energy, dU, of the parcel can occur only through an addition of heat, dH, to the parcel and/or by performing work, dW, on the parcel. This can be expressed mathematically as dU = dH + dW

(5.3a)

where each term may be considered to represent unit volume of the parcel. Experimental evidence suggests that the internal energy per unit volume of an ideal gas is proportional to its absolute temperature, so that dU = rcp dT for a constant pressure process, and dU = rcv dT for a constant volume process. Here, cp and cv are the speci®c heat capacities at constant pressure and volume, respectively. Their di€erence is equal to the speci®c gas constant, that is cp 7 cv = R (for dry air, cp = 1005 J K71 kg71 and R = 287.04 J K71 kg71). Expressing the work done on the air parcel in terms of pressure and volume, one can derive several di€erent expressions of the ®rst law of thermodynamics (Hess, 1959; Fleagle and Businger, 1980). Here, we give only the one involving changes in temperature and pressure rcpdT = dH + dP

(5.3b)

which implies that the parcel temperature may change due to an addition of heat and/or a change in pressure. If a parcel of air is lifted upward in the atmosphere, its pressure will decrease in response to decreasing pressure of its surroundings. This will lead to a decrease in the temperature and, possibly, to a decrease in the density of the parcel (due to expansion of the parcel) in accordance with Equation (5.2). If there is no exchange of heat between the parcel and the surrounding environment, the above process is called adiabatic. Vertical turbulent motions in the PBL, which carry air parcels up and down, are rapid enough to justify the assumption that changes in thermodynamic properties of such parcels are adiabatic (dH = 0). The rate of change of temperature of such a parcel with height can be calculated from Equations (5.1) and (5.3b) and is characterized by the adiabatic lapse rate (G) G =7(@T/@z)ad = g/cp

(5.4)

5.2 Basic Thermodynamic Relations and the Energy Equation

65

This is also the temperature lapse rate (rate of decrease of temperature with height) in an adiabatic atmosphere, which under dry or unsaturated conditions amounts to 0.0098 K m71 or nearly 10 K km71. The relationship between the changes of temperature and pressure in a parcel moving adiabatically, as well as those in an adiabatic atmosphere, are also given by Equations (5.2) and (5.3b), with dH = 0: dT/T = (R/cp)(dP/P)

(5.5)

integration of which gives the Poisson equation T = T0(P/P0)k

(5.6)

where T0 is the temperature corresponding to the reference pressure P0 and the exponent k = R/cp % 0.286. Equation (5.6) is used to relate the potential temperature Y, de®ned as the temperature which an air parcel would have if it were brought down to a pressure of 1000 mbar adiabatically from its initial state, to the actual temperature T as Y = T(1000/P)k

(5.7)

where P is in millibars. The potential temperature has the convenient property of being conserved during vertical movements of an air parcel, provided heat is not added or removed during such excursions. Then, the parcel may be identi®ed or labeled by its potential temperature. In an adiabatic atmosphere, potential temperature remains constant with height. For a nonadiabatic or diabatic atmosphere, it is easy to show from Equation (5.7) that, to a good approximation,   @Y Y @T @T ˆ ‡G % ‡G …5:8† @z T @z @z This approximation is particularly useful in the PBL where potential and actual temperatures, in absolute units, do not usually di€er by more than 10%. The above relationship is often used to express the di€erence in the potential temperatures between any two height levels as DY = DT + GDz

(5.9)

An integral version of the same relationship, Y 7 Y0 = T 7 T0 + Gz

(5.10)

66

5 Air Temperature and Humidity in the PBL

can be used to convert a temperature sounding (pro®le) into a potential temperature pro®le. The near-surface value Y0 can be calculated from Equation (5.7) if the surface pressure is known. A distinct advantage of using Equation (5.10) for calculating potential temperatures in the PBL is that one does not need to measure or estimate pressure at each height level. The above relations are applicable to both dry and moist atmospheres, so long as water vapor does not condense and one uses the appropriate values of thermodynamic properties for the moist air. In practice, it is found to be more convenient to use constant dry-air properties such as R and cp, but to modify some of the relations or variables for the moist air.

5.2.1 Moist unsaturated air There are more than a dozen variables used by meteorologists that directly or indirectly express the moisture content of the air (Hess, 1959, Chapter 4). Of these the most often used in micrometeorology is the speci®c humidity (Q), de®ned as the ratio, Mw/(Mw + Md), of the mass of water vapor to the mass of moist air containing the water vapor. In magnitude, it does not di€er signi®cantly from the mixing ratio, de®ned as the ratio, Mw/Md, of the mass of water vapor to the mass of dry air containing the vapor. Both are directly related to the water vapor pressure (e), which is the partial pressure exerted by water vapor and is a tiny fraction of the total pressure (P) anywhere in the PBL. To a good approximation, Q % mwe/mdP = 0.622e/P

(5.11)

where mw and md are the mean molecular masses of water vapor and dry air, respectively. The water vapor pressure is always less than the saturation vapor pressure (es), which is related to the temperature through the Clausius±Clapeyron equation (Hess, 1959, Chapter 4) des mw Le dT ˆ es R T2

…5:12†

where Le is the latent heat of vaporization or condensation. Integration of Equation (5.12) and using the empirical fact (condition) that at T = 273.2 K,

5.2 Basic Thermodynamic Relations and the Energy Equation

67

es = 6.11 mbar, yields ln

 es mw Le 1 ˆ 6:11 R 273:2

1 T

 …5:13†

in which es is in millibars. A plot of es versus T is called the saturation water vapor pressure curve and is given in many meteorological textbooks and monographs. Using Dalton's law of partial pressures (P = Pd + e), the equation of state for the moist air can be written as Pˆ

R R Md R  Mw rT ˆ T‡ T m md V mw V

or, after some algebra, Pˆ

  R md rT 1 ‡ md mw

  1 Q

…5:14†

After comparing Equation (5.14) to Equation (5.2) it becomes obvious that the factor in the brackets on the right-hand side of Equation (5.14) is the correction to be applied to the speci®c gas constant for dry air, R, in order to obtain the speci®c gas constant for the moist air. Instead of taking R as a humidity-dependent variable, in practice, the above correction is applied to the temperature in de®ning the virtual temperature   md Tv ˆ T 1 ‡ mw

  1 Q % T…1 ‡ 0:61Q†

…5:15†

so that the equation of state for the moist air is given by P = (R /md)rTv = RrTv

(5.16)

The virtual temperature is obviously the temperature which dry air would have if its pressure and density were equal to those of the moist air. Note that the virtual temperature is always greater than the actual temperature and the di€erence between the two, Tv 7 T % 0.61QT, may become as large as 7 K, which may occur over warm tropical oceans, and is usually less than 2 K over mid-latitude land surfaces. For moist air one can de®ne the virtual potential temperature, Yv, similar to that for dry air and use Equations (5.7) and (5.8) with Y and T replaced by Yv and Tv, respectively.

68

5 Air Temperature and Humidity in the PBL

The mixing ratio of water vapor in air being always very small (50.04), the speci®c heat capacity is not signi®cantly di€erent from that of dry air. Therefore, changes in state variables of moving air parcels in an unsaturated atmosphere are not signi®cantly di€erent from those in a dry atmosphere, so long as the processes remain adiabatic. In particular, the adiabatic lapse rate given by Equation (5.4) is also applicable to most unsaturated air parcels moving in a moist environment, provided T in Equation (5.4) is replaced by Tv.

5.2.2 Saturated air In saturated air, a part of the water vapor may condense and, consequently, the latent heat of condensation may be released. If condensation products (e.g., water droplets and ice crystals) remain suspended in an upward-moving parcel of saturated air, the saturated lapse rate, or the rate at which the saturated air cools as it expands with the increase in height, must be lower than the dry adiabatic lapse rate G, because of the addition of the latent heat of condensation in the former. If there is no exchange of heat between the parcel and the surrounding environment, such a process is called moist adiabatic. From the ®rst law of thermodynamics for the moist adiabatic process and the equation of state, the moist adiabatic or saturated lapse rate is given by (Hess, 1959):  Gs ˆ

@T @z

  dQs ˆ g cp ‡ Le dT s



1

…5:17†

where dQs/dT is the slope of the saturation-speci®c humidity versus temperature curve, as given by Equations (5.12) and (5.13). Unlike the dry adiabatic lapse rate, however, the saturated adiabatic lapse rate is quite sensitive to temperature. For example, at T = 273.2 K, Gs = 0.0069 K m71 and at T = 303 K, Gs = 0.0036 K m71, both at the reference sea-level pressure of 1000 mbar. In the real atmospheric situation, some of the condensation products are likely to fall out of an upward-moving parcel, while others may remain suspended as cloud particles in the parcel. As a result of these changes in mass and composition of the parcel, some exchange of heat with the surrounding air must occur. Therefore the processes within the parcel are not truly adiabatic, but are called pseudoadiabatic. The mass of condensation products being only a tiny fraction of the total mass of parcel, however, the pseudoadiabatic lapse rate is not much di€erent from the moist adiabatic lapse rate. The di€erence between the dry adiabatic and moist adiabatic lapse rates accounts for some interesting orographic phenomena, such as the formation of

5.2 Basic Thermodynamic Relations and the Energy Equation

69

clouds and possible precipitation on the upwind slopes of mountains and warm and dry downslope winds (e.g., Chinook and Foen winds) near the lee side base (Rosenberg et al., 1983, Chapter 3). Example Problem 1 A moist air parcel with a temperature of 208C and speci®c humidity of 10 g kg71 is lifted adiabatically from the upwind base of a mountain, where the pressure is 1000 mb, to its top, at 3000 m above the base, and is then brought down to the base on the other side of the mountain. Using the appropriate thermodynamic relations or diagrams, estimate the following parameters: (a) The height zs above the base where the air parcel would become saturated and the saturated adiabatic lapse rate at this level. (b) The temperature of the parcel at the mountain top. (c) The temperature of the parcel at the downwind base. Solution (a) First, we determine the temperature T(zs) at which the parcel would become saturated from Equation (5.13)    e  1 1 s ln ˆ 5404 6:11 273:2 T…zs † where, from Equation (5.11), es = QsP/0.622 % 0.0161P(zs) The pressure and temperature of the parcel at the height zs are related through the Poisson equation [Equation (5.6)] as T(zs) = T0[P(zs)/P0]0.286 The above expressions form a closed loop which can be solved, using an iterative procedure, to determine the variables at zs as follows: T(zs) = 285.9 K; P(zs) = 917 mb; es = 14.73 mb Then the height zs is given by zs = (293.2 7 285.9)/0.0098 % 745 m Alternatively, zs, T(zs), and P(zs) can also be determined from an appropriate thermodynamic diagram.

70

5 Air Temperature and Humidity in the PBL The saturated adiabatic lapse rate can be determined from Equation (5.17) in which dQs mw Le es ˆ 0:622 % 6:48  10 dT R PT2

4

K

1

Then, Gs = 9.816/(1010 + 2.45 6 648) % 0.0038 K m71 (b) Assuming a constant saturated adiabatic lapse rate up to the top of the mountain at zt = 3000 m, the temperature of the parcel there is given by T(zt) = T(zs) 7 Gs(zt 7zs) % 277.4 K However, this is only an approximation, because in reality dQs/dT is expected to vary with height. A more accurate estimate of the temperature at the mountain top can be obtained by following the moist adiabatic for T = 286 from the 745 m height level to 3000 m on a suitable thermodynamic diagram or chart. (c) Assuming that the parcel remained unsaturated during its descent on the other side of the mountain, the temperature at the downwind base is given by Tb = 277.4 + 0.0098 6 3000 = 306.8 K which is 13.6 K higher than the air temperature at the upwind base. This exercise shows that, in going up and down a high mountain, air temperature can increase substantially, provided that the lifting condensation level is well below the mountain top.

5.2.3 Thermodynamic energy equation Using the principle of the conservation of energy within an elementary volume of air, one can derive the following di€erential equation for the variation of temperature (Kundu, 1990, Chapter 4): DT ˆ ah r2 T ‡ SH Dt

…5:18†

5.3 Local and Nonlocal Concepts of Static Stability

71

where ah is thermal di€usivity and SH is the source term representing the rate of change of temperature within the elementary volume. Here, the total derivative DT/Dt represents the sum of the local change of temperature with time and advective changes in the same due to ¯ow, while the H2 (Laplacian) operator has the usual meaning. In deriving the simpli®ed Equation (5.18), the so-called Boussinesq's approximations have been used and any change in temperature due to radiative ¯ux divergence has been neglected. A similar equation can also be written for potential temperature.

5.3 Local and Nonlocal Concepts of Static Stability The variations of temperature and humidity with height in the PBL lead to density strati®cation with the consequence that an upward- or downwardmoving parcel of air will ®nd itself in an environment whose density will, in general, di€er from that of the parcel, after accounting for the adiabatic cooling or warming of the parcel. In the presence of gravity this density di€erence leads to the application of a buoyancy force on the parcel which would accelerate or decelerate its vertical movement. If the vertical motion of the parcel is enhanced, i.e., the parcel is accelerated farther away from its equilibrium position by the buoyancy force, the environment is called statically unstable. On the other hand, if the parcel is decelerated and is moved back to its equilibrium position, the atmosphere is called stable or stably strati®ed. When the atmosphere exerts no buoyancy force on the parcel at all, it is considered neutral. In general the buoyancy force or acceleration exerted on the parcel may vary with height and so does the static stability. One can derive an expression for the buoyant acceleration, ab, on a parcel of air by using the Archimedes principle and determining the net upward force due to buoyancy

ab ˆ g

r

rp

! …5:19†

rp

in which the subscript p refers to the parcel. Further, using the equation of state for the moist air, Equation (5.19) can also be written as  ab ˆ

g

Tv

Tvp Tv

 …5:20†

72

5 Air Temperature and Humidity in the PBL

5.3.1 Local static stability Expressions (5.19) and (5.20) for the buoyant acceleration on a parcel in a strati®ed environment are valid irrespective of the origin or initial position of the parcel. An alternative, but approximate, expression for ab in terms of the local gradient of virtual temperature or potential temperature is often given in the literature as ab %

  g @Tv ‡ G Dz ˆ Tv @z

g @Yv Dz Tv @z

…5:21†

This expression can be derived from Equation (5.20) by assuming that any parcel displacement, Dz, from its equilibrium position is small and by expanding Tv and Tvp in the Taylor series around their equilibrium value Tve. Equation (5.21) gives a criterion, as well as a quantitative measure, of static stability of the atmosphere in terms of @Tv/@z or @Yv/@z. In particular, the static stability parameter is de®ned as s = (g/Tv)(@Yv/@z)

(5.22)

Thus, qualitatively, the static stability is often divided into three broad categories: 1. Unstable, when s 5 0, @Yv/@z 5 0, or @Tv/@z 57G. 2. Neutral, when s = 0, @Yv/@z = 0, or @Tv/@z =7G. 3. Stable, when s 4 0, @Yv/@z 4 0, or @Tv/@z 47G. The magnitude of s provides a quantitative measure of static stability. It is quite obvious from Equation (5.21) that vertical movements of parcels up or down are generally enhanced in an unstable atmosphere, while they are suppressed in a stably strati®ed environment. Note that, for an upward-moving parcel, ab 4 0 in a statically unstable environment and ab 5 0 in a stable environment. On the basis of virtual temperature gradient or lapse rate (LR), relative to the adiabatic lapse rate G, atmospheric layers are variously characterized as follows: 1. 2. 3. 4. 5.

Superadiabatic, when LR 4 G. Adiabatic, when LR = G. Subadiabatic, when 0 5 LR 5 G. Isothermal, when @Tv/@z = 0. Inversion, when @Tv/@z 4 0.

5.3 Local and Nonlocal Concepts of Static Stability

73

Figure 5.1 Schematic of the various stability categories on the basis of virtual temperature gradient.

Figure 5.1 gives a schematic of these static stability categories in the lower part of the PBL. Here, the surface temperature is assumed to be the same and virtual temperature pro®les are assumed to be linear for convenience only; actual pro®les are usually curvilinear, as will be shown later. This traditional view of static stability, based on the local gradient of virtual temperature or potential temperature has some serious limitations and is not very satisfactory, especially when static stability is used as a measure of turbulent mixing and di€usion. For example, it is well known that in the bulk of the convective boundary layer (CBL), @Yv/@z % 0, or is slightly positive. Its characterization as a neutral or slightly stable layer, according to the traditional de®nition of static stability based on the local gradient, would be very misleading because the convective mixed layer (the middle part of the CBL where @Yv/@z % 0) has all the attributes of an unstable boundary layer, such as upward heat ¯ux, vigorous mixing, and large thickness. It would be correctly recognized as an unstable layer if one considers the positive buoyancy forces or accelerations that would act in all the warm air parcels originating in the superadiabatic layer near the surface. For these parcels, experiencing large displacements, before they arrive in the mixed layer, Equation (5.21) is not valid and the local static stability parameter becomes irrelevant. Thus, s is not a good

74

5 Air Temperature and Humidity in the PBL

measure of buoyant accelerations or forces on air parcels undergoing large displacements from their equilibrium positions. The di€erences in densities or virtual temperatures between parcels and the environment are the simplest and most direct measures of buoyancy forces and should be considered in any stability characterization of the environment. The original criterion for determining static stability in terms of buoyancy forces on parcels coming from all possible initial positions appears to be basically sound. It is only when one considers the alternative criterion in terms of the local lapse rate, @Yv/@z, or s that ambiguities arise in certain situations. 5.3.2 Nonlocal static stability In order to remove the ambiguities associated with the concept of local static stability, Stull (1988, Chapter 5; 1991) has proposed a nonlocal characterization of stability. For this, an environmental sounding of thermodynamic variables should be taken over a deep layer between the surface and the level where vertical parcel movements are likely to become insigni®cant (e.g., strong inversion or the tropopause). To determine the stability of the various layers, air parcels should be displaced up and down from all possible starting points within the whole domain. In practice, one needs to consider parcels starting from only the relative minima and maxima of the virtual potential temperature sounding or pro®le. Air parcel movement up or down from the initial position should be based on the parcel buoyancy and not on the local lapse rate. The parcel buoyancy at any level is determined by the di€erence in virtual temperatures of the parcel and the environment at that level. Buoyancy forces warm air parcels to rise and cold parcels to sink. Displaced air parcels that would continue to move farther away from their starting level should be tracked all the way to the level where they would become neutrally buoyant (Tvp = Tv, or Yvp = Yv). After parcel movements from all the salient levels (minima and maxima in the Yv pro®le) have been tracked, nonlocal static stability of di€erent portions (layers) of the sounding domain should be determined in the following order (Stull, 1991): 1. Unstable: Those regions in which parcels can enter and transit under their own buoyancy. Individual air parcels need not traverse the whole unstable region. Many parcels would traverse only a part of the unstable region or layer. 2. Stable: Those regions of subadiabatic lapse rate that are not unstable. 3. Neutral: Those regions of adiabatic lapse rate that are not unstable. 4. Unknown: Those top or bottom portions of the sounding that are apparently stable or neutral, but that do not end at a material surface, such as the

5.3 Local and Nonlocal Concepts of Static Stability

75

ground surface, strong inversion, or the tropopause. Above or below the known sounding region might be a cold or warm layer, respectively, that could provide a source of buoyant air parcels. The above method or procedure for determining the nonlocal stability is illustrated in Figure 5.2 for the various hypothetical, but practically feasible, soundings (Yv pro®les). Note that in some of these cases, especially those with unstable regions, the traditional characterizations of stability based on the local lapse rate or @Yv/@z would be quite di€erent from the broader, nonlocal characterization used here. The latter are based on the buoyant movements (indicated by vertical dashed lines and arrows in Figure 5.2) of air parcels, some of which are undergoing large displacements. When compared with traditional local static stability measures, the nonlocal characterization of stability is more consistent with the empirical evidence that unstable and convective boundary layers have strong and ecient mixing, while the stable boundary layer and other inversion layers have relatively weak mixing associated with them. The former is recommended as the preferred choice for

Figure 5.2 Nonlocal stability characterization for the various hypothetical virtual potential temperature pro®les. [After Stull (1988).]

76

5 Air Temperature and Humidity in the PBL

applications in micrometeorology and air pollution meteorology (Arya, 1999, Chapter 2). Example Problem 2 The following measurements were made from the research vessel FLIP during the 1969 Barbados Oceanographic and Meteorological Experiment (BOMEX): Height above sea-surface (m): Air temperature (8C): Speci®c humidity (g kg71):

2.1 27.74 18.02

3.6 27.72 17.83

6.4 27.69 17.54

11.4 27.62 17.29

(a) Calculate the virtual temperature at each level and its di€erence from the actual temperature. (b) Calculate and compare the gradients of potential temperature and virtual potential temperature between adjacent measurement levels. (c) Calculate the local static stability parameter, s, and the percentage contribution of speci®c humidity gradient to the same at di€erent height levels. Solution (a) Using Tv = T(1 + 0.61Q) and Tv 7 T = 0.61QT, where T and Tv are in absolute units (K) and Q is dimensionless, we obtain the following: z (m): 2.1 304.25 Tv (K): 3.31 (Tv 7 T ) (K):

3.6 304.19 3.21

6.4 304.11 3.22

11.4 303.99 3.17

(b) Estimating the gradients between adjacent levels as @T DT % ; @z Dz

@Tv DTv % @z Dz

and using the approximate relations @Y @T % ‡ G; @z @z

@Yv @Tv % ‡G @z @z

We can estimate and compare the above gradients as follows: Mean height (m): 2.85 @Y/@z (K m71): 70.0035 @Yv/@z (K m71): 70.0302

5.00 70.0009 70.0188

8.90 70.0042 70.0142

5.4 Mixed Layers, Mixing Layers and Inversions

77

Note that magnitudes of @Yv/@z over the tropical ocean are much larger than those of @Y/@z. (c) Using the de®nition sˆ

g @Yv ; Tv @z

it can easily be calculated. For determining the relative contribution of the speci®c humidity gradient to s, we can write sˆ

    g @Tv g @T @Q ‡G ˆ ‡ 0:61T ‡G Tv @z Tv @z @z   g @Y @Q ‡ 0:61T ˆ Tv @z @z

The second term represents the contribution of the speci®c humidity gradient to s. The relative contribution of the same is given by @Q 0:61T @z



@Yv ˆ1 @z

@Y=@z @Yv =@z

Mean height (m): 2.85 5.00 8.90 79.74 6 1074 76.02 6 1074 74.54 6 1074 Static stability (s72): % contribution of @Q=@z: 88.4 95.2 70.4 This example illustrates that it is extremely important to include the speci®c humidity gradients in any estimates of static stability over water surfaces. The stability based on temperature alone would be characterized as near-neutral, while it is signi®cantly unstable, because of the stronger negative humidity gradients. 5.4 Mixed Layers, Mixing Layers and Inversions In moderate to extremely unstable conditions, typically encountered during midday and afternoon periods over land, mixing within the bulk of the boundary layer is intense enough to cause conservable scalars, such as Y and Yv, to be distributed uniformly, independent of height. The layer over which this occurs is called the mixed layer, which is an adiabatic layer overlying a superadiabatic surface layer. To a lesser extent, the speci®c humidity, momentum, and concentrations of pollutants from distant sources also have uniform

78

5 Air Temperature and Humidity in the PBL

distributions in the mixed layer. Mixed layers are found to occur most persistently over the tropical and subtropical oceans. Sometimes, the term `mixing layer' is used to designate a layer of the atmosphere in which there is signi®cant mixing, even though the conservable properties may not be uniformly mixed. In this sense, the turbulent PBL is also a mixing layer, a large part of which may become well mixed (mixed layer) during unstable conditions. Mixing layers are also found to occur sporadically or intermittently in the free atmosphere above the PBL, whenever there are suitable conditions for the breaking of internal gravity waves and the generation of turbulence in an otherwise smooth, streamlined ¯ow. The term `inversion' in common usage applies to an atmospheric condition when air temperature increases with height. Micrometeorologists sometimes refer to inversion as any stable atmospheric layer in which potential temperature (more appropriately, Yv) increases with height. The strength of inversion is expressed in terms of the layer-averaged @Yv/@z or the di€erence DYv across the given depth of the inversion layer. Inversion layers are variously classi®ed by meteorologists according to their location (e.g., surface and elevated inversions), the time (e.g., nocturnal inversion), and the mechanism of formation (e.g., radiation, evaporation, advection, subsidence, frontal and sea-breeze inversions). Because vertical motion and mixing are considerably inhibited in inversions, a knowledge of the frequency of occurrence of inversions and their physical characteristics (e.g., location, depth, and strength of inversion) is very important to air pollution meteorologists. Pollutants released within an inversion layer often travel long distances without much mixing and spreading. These are then fumigated to ground level as soon as the inversion breaks up to the level of the ribbonlike thin plume, resulting in large ground-level concentrations. Low-level and ground inversions act as lids, preventing the downward di€usion or spread of pollutants from elevated sources. Similarly, elevated inversions put an e€ective cap on the upward spreading of pollutants from low-level sources. An inversion layer usually caps the daytime unstable or convective boundary layer, con®ning the pollutants largely to the PBL. Consequently, the PBL depth is commonly referred to as the mixing depth in the air quality literature. The most persistent and strongest surface inversions are found to occur over the Antarctica and Arctic regions; the tradewind inversions are probably the most persistent elevated inversions capping the PBL. 5.5 Vertical Temperature and Humidity Pro®les A typical sequence of observed potential temperature pro®les at 3 h intervals during the course of a day is shown in Figure 5.3. These were obtained from

5.5 Vertical Temperature and Humidity Pro®les

79

Figure 5.3 Diurnal variations of potential temperature pro®les and inversion heights during (a) day 33 and (b) days 33±34 of the Wangara Experiment. (c) Curve A, inversion base height in the daytime CBL. Curve B, surface inversion height in the nighttime SBL. [After Deardor€ (1978).]

radiosonde measurements during the 1967 Wangara Experiment near Hay, New South Wales, Australia, on a day when there were clear skies, very little horizontal advections of heat and moisture, and no frontal activity within 1000 km. Note that, before sunrise and at the time of the minimum in the near-surface temperature, the Y pro®le is characterized by nocturnal inversion, which is produced as a result of radiative cooling of the surface. The nocturnal boundary layer is stably strati®ed, so that vertical turbulent exchanges are greatly suppressed. It is generally referred to as the stable boundary layer (SBL). Shortly after sunrise, surface heating leads to an upward exchange of sensible

80

5 Air Temperature and Humidity in the PBL

heat and subsequent warming of the lowest layer due to heat ¯ux convergence. This process progressively erodes the nocturnal inversion from below and replaces it with an unstable or convective boundary layer (CBL) whose depth h grows with time (see Figure 5.3c). The rate of growth of h usually attains maximum a few hours after sunrise when all the surface-based inversion has been eroded, and slows down considerably in the late afternoon. The daytime unstable or convective boundary layer has a three-layered structure. The potential temperature decreases with height within the shallow surface layer. In this layer, turbulence is generated by both wind shear and buoyancy. The surface layer is followed by the mixed layer in which potential temperature remains more or less uniform. The mixed layer comprises the bulk of the CBL. This layer is dominated by buoyancy-generated turbulence, except during morning and evening transition hours when shear e€ects also become important. Above the mixed layer is a shallow transition layer in which potential temperature increases with height. In this stable layer, turbulence decreases rapidly with height. From the temperature sounding alone, the top of the PBL, including the transition layer, is not as distinct as the top of the mixed layer which is also the base of the inversion. Therefore, the height of inversion (zi) is often used as an approximation for the PBL height (h), although the former can be substantially (10±30%) smaller than the latter. Just before sunset, there is net radiative loss of energy from the surface and, consequently, an inversion forms at the surface. The nocturnal inversion deepens in the early evening hours and sometimes throughout the night, as a result of a radiative and sensible heat ¯ux divergence. The surface inversion height (hi) may not correspond to the height, h, of the stable boundary layer (SBL) based on signi®cant turbulence activity. The inversion height is more easily detected from temperature soundings and is depicted in Figure 5.3c. The PBL height can be directly measured from remote-sensing instruments such as sodar and lidar or indirectly estimated from both temperature and wind pro®les. Such measurements indicate that, unlike the surface inversion height, the PBL height (h) generally decreases with time during early evening hours and attains a constant value which is usually less than hi. Sometimes, the PBL height oscillates around some average value throughout the night. The SBL is often capped by the remnant of the previous afternoon's mixed layer. Mixing processes in this so-called residual layer are likely to be very weak. Strong wind shears may generate some intermittent and sporadic turbulence, which is not connected to the SBL. Observed vertical pro®les of speci®c humidity for the same Wangara day 33 are shown in Figure 5.4. The surface during this period of drought was rather dry, with little or no growth of vegetation (predominantly dry grass, legume, and cottonbush). In the absence of any signi®cant evapotranspiration from the surface, the speci®c humidity pro®les are nearly uniform in the daytime PBL,

5.5 Vertical Temperature and Humidity Pro®les

81

Figure 5.4 Diurnal variation of speci®c humidity pro®les during day 33 of the Wangara Experiment. [From Andre et al. (1978).]

with the value of Q changing largely in response to the evolution of the mixed layer. In other situations where there is dry air aloft, but plenty of moisture available for evaporation at the surface, the evolution of Q pro®les might be quite di€erent from those shown in Figure 5.4. In response to the increase in the rate of evaporation or evapotranspiration during the morning and early afternoon hours, Q may actually increase and attain its maximum value sometime in the afternoon and, then, decrease thereafter, as the evaporation rate decreases. The evaporation from the surface, although strongest by day, may continue at a reduced rate throughout the night. Under certain conditions, when winds are light and the surface temperature falls below the dew point temperature of the moist air near the surface, the water vapor is likely to condense and deposit on the surface in the form of dew. The resulting inversion in the Q pro®le leads to a downward ¯ux of water vapor and further deposition. Temperature and humidity pro®les over open oceans do not show any signi®cant diurnal variation when large-scale weather conditions remain unchanged, because the sea-surface temperature changes very little (typically, less than 0.58C), if at all, between day and night. Figure 5.5 shows the observed pro®les of T and Q from two research vessels (with a separation distance of about 750 km) during the 1969 Atlantic Tradewind Experiment (ATEX). Here, only the averaged pro®les for the two periods designated as `undisturbed' (February 7±12) and `disturbed' (February 13±17) weather are given. During the undisturbed period, the temperature and humidity structure of the PBL is characterized by a mixed layer (depth % 600 m), followed by a shallow

82

5 Air Temperature and Humidity in the PBL

Figure 5.5 Observed mean vertical pro®les of temperature and speci®c humidity at two research vessels during ATEX. First (`undisturbed') period: solid lines; second (`disturbed') period: dashed lines. The dotted lines indicate the dry adiabatic and saturated adiabatic lapse rates. [After Augstein et al. Copyright # (1974) by D. Reidel Publishing Company. Reprinted by permission.]

isothermal transition layer (depth % 50 m) and a deep cloud layer extending to the base of the trade wind inversion. The same features can also be seen during the disturbed period, but with larger spatial di€erences. At the other extreme, Figure 5.6 shows the measured mean temperature pro®les in the surface inversion layer at the Plateau Station, Antarctica (79815'S, 40830'E), in three di€erent seasons (periods) and for di€erent stability classes (identi®ed here as (1) through (8) in the order of increasing stability and de®ned from both the temperature and wind data). These include some of the most stable conditions ever observed on the earth's surface. 5.6 Diurnal Variations As a result of the diurnal variations of net radiation and other energy ¯uxes at the surface, there are large diurnal variations in air temperature and speci®c humidity in the PBL over land surfaces. The diurnal variations of the same over large lakes and oceans are much smaller and often negligible. The largest diurnal changes in air temperature are experienced over desert areas where strong large-scale subsidence limits the growth of the PBL and the air is rapidly heated during the daytime and is rapidly cooled at night. The mean diurnal range of screen-height temperatures is typically 208C, with mean maximum temperatures of about 458C during the summer months (Deacon, 1969).

Figure 5.6 Observed mean temperature pro®les in the surface inversion layer at Plateau Station, Antarctica, for three di€erent periods grouped under di€erent stability classes. (a) Sunlight periods, 1967; (b) transitional periods, 1967; (c) dark season, 1967. [After Lettau et al. (1977).]

84

5 Air Temperature and Humidity in the PBL

The diurnal range of temperatures is reduced by the presence of vegetation and the availability of moisture for evaporation. Even more dramatic reductions occur in the presence of cloud cover, smoke, haze, and strong winds. The diurnal range of temperatures decreases rapidly with height in the PBL and virtually disappears at the maximum height of the PBL or inversion base, which is reached in the late afternoon. Some examples of observed diurnal variations of air temperature at di€erent heights and under di€erent cloud covers and wind conditions are given in Figure 5.7.

Figure 5.7 Diurnal variation of air temperature at three heights (solid lines: 1.2 m; dashed lines: 7 m; dotted lines: 17 m) over downland in Southern England for various combinations of clear and overcast days in June and December. Wind speeds at 12 m height are also indicated. [From Deacon (1969); after Johnson (1929).]

5.7 Applications

85

Figure 5.8 Diurnal variation of water vapor pressure at three heights at Quickborn, Germany, on clear May days. [From Deacon (1969).]

The diurnal variation of speci®c humidity depends on the diurnal course of evapotranspiration and condensation, surface temperatures, mean winds, turbulence, and the PBL height. Large diurnal changes in surface temperature generally lead to large variations of speci®c humidity, due to the intimate relationship between the saturation vapor pressure and temperature. Figure 5.8 illustrates the diurnal course of water vapor pressure at three di€erent heights. Here the secondary minimum of vapor pressure in the afternoon is believed to be caused by further deepening of the mixed layer after evapotranspiration has reached its maximum value. This phenomenon may be even more marked at other places and times. 5.7 Applications The knowledge of air temperature and humidity in the PBL is useful and may have applications in the following areas: . Determining the static stability of the lower atmosphere and identifying mixed layers and inversions. . Determining the PBL height, as well as the depth and strength of surface or elevated inversion. . Predicting air temperature and humidity near the ground and possible fog and frost conditions. . Determining the sensible heat ¯ux and the rate of evaporation from the surface. . Calculating atmospheric radiation.

86

5 Air Temperature and Humidity in the PBL

. Estimating di€usion of pollutants from surface and elevated sources and possible fumigation. . Designing heating and air-conditioning systems.

Problems and Exercises 1. How do the mechanisms of heat transfer in the atmosphere di€er from those in a subsurface medium, such as soil and rock? 2. (a) Derive the following equation for the variation of air temperature with time, in the absence of any horizontal and vertical advections of heat: rcp(@T/@t) = (@RN/@z) 7 (@H/@z)

(b) Indicate how the above equation may be used to estimate the sensible heat ¯ux at the surface, H0, during the daytime when the rate of warming is nearly constant through the depth of the PBL. (c) Temperature measurements at the top of a 20 m tower indicated a rate of warming of 1.8 8C h71 during the mid-day period when the PBL height was estimated to be 850 m from a minisonde sounding. Estimate the surface heat ¯ux at the time of observations. 3. (a) Derive an expression for the adiabatic lapse rate in a moist, unsaturated atmosphere and discuss the e€ect of moisture on the same. (b) Explain why the saturated adiabatic lapse rate must be smaller than the dry diabatic lapse rate. 4. (a) Derive the equation of state for the moist air and hence de®ne the virtual temperature Tv. (b) Show that, to a good approximation, @Tv/@z = (@T/@z) + 0.61T(@Q/@z) 5. (a) Show that the buoyant acceleration on a parcel of air in a thermally strati®ed environment is given by ab = 7(g/Tv)(Tv 7 Tvp)

Problems and Exercises

87

(b) Considering a small vertical displacement, Dz, from the equilibrium position of the parcel, show that ab % 7(g/Tv)(@Yv/@z)Dz 6. A moist air parcel at a temperature of 308C and speci®c humidity of 25 g kg71 is lifted adiabatically from the upwind base of a tropical mountain island, where the pressure is 1000 mb, to the top, at 2000 m above the base, and is then brought down to the base on the other side of the mountain. Using appropriate equations or tables, calculate the following parameters: (a) The height above the base where the air parcel would become saturated and the saturated adiabatic lapse rate at this level. (b) The temperature of the parcel at the top of the mountain. (c) The temperature of the parcel at the downwind base, assuming that the parcel remained unsaturated during its entire descent. 7. The following pressure, temperature and speci®c humidity measurements were made from radiosonde soundings on day 11 of the Wangara PBL experiment: Height (m)

Pressure (mb)

Temperature (8C)

Speci®c humidity (g kg71)

Surface (1.5) 50 100 150 200 300 400 500 550 600 650 700 750 800 850 900 1000

1013 1007 1001 995 989 977 965 954 948 942 936 931 925 919 914 908 897

11.8 10.9 10.1 9.6 9.2 8.4 7.4 6.4 6.0 5.6 5.1 4.9 4.9 4.9 5.0 5.0 4.8

6.5 6.1 5.8 5.6 5.5 5.3 5.1 4.9 4.9 4.8 4.7 4.6 4.5 4.5 4.5 4.5 4.4

(a) Calculate and plot virtual temperature as a function of height. (b) Calculate and plot virtual potential temperature as a function of height.

88

5 Air Temperature and Humidity in the PBL

(c) Characterize the various layers on the basis of local and nonlocal stability and discuss the di€erences between the two. (d) Estimate the height of the inversion base (zi), as well as the PBL height (h). Discuss the sensitivity of the latter to the near-surface temperature. 8. Discuss the importance of speci®c humidity gradient in the determination of the static stability parameter s over di€erent types of surfaces. Under what conditions can one ignore speci®c humidity in the determination of static stability?

Chapter 6

Wind Distribution in the PBL

6.1 Factors In¯uencing Wind Distribution The magnitude and direction of near-surface winds and their variations with height in the PBL are of considerable interest to micrometeorologists. The following factors in¯uence the wind distribution in the PBL: . Large-scale horizontal pressure and temperature gradients in the lower atmosphere, which drive the PBL ¯ow. . The surface roughness characteristics, which determine the surface drag and momentum exchange in the lower part of the PBL. . The earth's rotation, which makes wind turn with height. . The diurnal cycle of heating and cooling of the surface, which determines the thermal strati®cation of the PBL. . The PBL depth, which determines wind shears in the PBL. . Entrainment of the free atmospheric air into the PBL, which determines the momentum, heat, and moisture exchanges at the top of the PBL, as well as the PBL height. . Horizontal advections of momentum and heat, which a€ect both the wind and temperature distributions in the PBL. . Large-scale horizontal convergence or divergence and the resulting mean vertical motion at the top of the PBL. . Presence of clouds and precipitation in the PBL, which in¯uence its thermal strati®cation. . Surface topographical features, which give rise to local or mesoscale circulations. Some of these factors will be discussed in more detail in the following text, while others are considered outside the scope of this introductory text.

89

90

6 Wind Distribution in the PBL

6.2 Geostrophic and Thermal Winds The PBL is essentially driven by large-scale atmospheric motions which are set up in response to spatial variations of air pressure and temperature. The geostrophic winds are related to or de®ned in terms of the pressure gradients as Ug ˆ

1 @P 1 @P ; Vg ˆ rf @y rf @x

…6:1†

in which Ug and Vg are the components of geostrophic wind vector G in the x and y directions, respectively, and f is the Coriolis parameter, which is related to the rotational speed of the earth (O) and latitude (f) as f = 2O sin f

(6.2)

The geostrophic winds are the winds which would occur as a result of the simple geostrophic balance between the pressure gradient and Coriolis (rotational) forces in a frictionless atmosphere with no advective and local accelerations. Equation (6.1) implies that G must be parallel to the isobars with low pressure to the left in the northern hemisphere ( f 4 0) (see Figure 6.1) and low pressure to the right in the southern hemisphere ( f 5 0). The absence of advective accelerations further implies that, locally, isobars are equispaced, parallel, straight lines. The geostrophic balance is often assumed to occur, i.e., U = Ug and V = Vg, at the top of and outside the PBL. Within the PBL, however, actual

Figure 6.1 Schematic of relationship between geostrophic winds and isobars for the northern hemisphere.

6.2 Geostrophic and Thermal Winds

91

winds di€er from the geostrophic winds because of the surface friction and the vertical exchange of momentum. The horizontal pressure gradients or geostrophic winds may vary with height in response to horizontal temperature gradients. The vertical gradients of geostrophic winds, i.e., geostrophic wind shears, are given by the thermal wind equations @Ug ˆ @z

g @T Ug @T ‡ % f T @y T @z

g @T f T @y

@Vg g @T Vg @T g @T ˆ ‡ % f T @x f T @x @z T @z

…6:3†

which can be derived from Equation (6.1) in conjunction with the hydrostatic equation and the equation of state (Hess, 1959, Chapter 12). The approximations on the right-hand side of Equation (6.3) ignore the stability-dependent terms, which may account for no more than 10% variation in geostrophic winds per kilometer of height. The approximate thermal wind equations imply that the geostrophic wind shear vector @G/@z must be parallel to the isotherms with colder air to the left in the northern hemisphere (Figure 6.2). A part of the geostrophic wind shear is due to the normal (climatological) decrease of temperature in going toward the poles, while a substantial part may be due to local temperature gradients created by surface topographical features (e.g., land±sea, urban±rural, and mountain±valley contrasts) and/or the synoptic weather situation. Note that a horizontal temperature gradient of only 18C per 100 km will cause a geostrophic wind shear of about 3.3 m s71 km71 or 0.0033 s71 in middle latitudes.

Figure 6.2 Schematic of relationship between thermal winds or geostrophic shears and isotherms for the northern hemisphere.

92

6 Wind Distribution in the PBL

The changes in geostrophic wind components over a given layer depth, Dz, can be calculated from the ®nite-di€erence form of Equation (6.3): DUg ˆ

Dzg @T Dzg @T ; DVg ˆ f T @y f T @x

…6:4†

Horizontal temperature gradients or geostrophic shears are generally assumed to be constant (independent of height) in the PBL. It follows from Equation (6.3) or (6.4) that geostrophic winds must vary with height in the presence of ®nite temperature gradients. Figure 6.3 illustrates for the northern hemisphere the relationship between the geostrophic winds at the surface and the top of the PBL (taking Dz = h) for di€erent orientations of the surface geostrophic wind

Figure 6.3 Schematic of the relationship between the geostrophic winds at the surface and the top of the PBL for di€erent orientations of surface isobars or G0 and surface isotherms: (a) parallel geostrophic ¯ow around a `cold low' or a `warm high'; (b) parallel ¯ow around a `cold high', or a `warm low'; (c) veering geostrophic ¯ow in warm advection; (d) backing ¯ow in cold advection.

6.2 Geostrophic and Thermal Winds

93

G0 with respect to the isotherms. Again, isobars and isotherms are taken as parallel straight lines for the ideal case of no advective acceleration. Note that when G0 is parallel to the isotherms (no temperature advection), the e€ect of the horizontal temperature gradient is to increase or decrease the magnitude of geostrophic winds, without causing any change in the direction. This will be the rare situation when isobars are parallel to the isotherms throughout the layer. In general, however, the surface isobars are not parallel to the isotherms and the surface geostrophic wind has a component from cold toward warm temperatures or vice versa. Such a ¯ow tends to change the vertical temperature distribution due to horizontal advection of cold or warm air. Figure 6.3 shows that both the magnitude and the direction of the geostrophic wind may change with height in the presence of heat advection. The geostrophic wind turns cyclonically (backs) with height in the case of cold air advection, and it turns anticyclonically (veers) with height in the case of warm advection. Actual winds within the PBL may also be expected to back or veer with height in response to the backing and veering of geostrophic winds. Equations (6.1) and (6.4) express the geostrophic and thermal winds in terms of horizontal pressure and temperature gradients on a horizontal (constantlevel) surface. These can be used with the information provided on a surface chart depicting isobars and isotherms. The geostrophic and thermal winds in meteorology are often expressed in terms of the heights (zp) of constant-pressure surfaces which are represented on upper-air (e.g., 850 mb, 700 mb, etc.) charts. On any constant-pressure surface, the geostrophic and thermal winds can be expressed as Ug ˆ

DUg ˆ

g @zp g @zp ; Vg ˆ f @y f @x

g @…Dzp † g @…Dzp † ; DVg ˆ f @y f @x

…6:5†

…6:6†

where Dzp is the thickness of the layer between the p1 and p2 constant-pressure surfaces and DUg and DVg are the corresponding thermal wind components. The hypsometric equation can be used to express Dzp in terms of p1 and p2, and the average temperature of the layer. Equation (6.5) implies that the geostrophic wind is proportional to the slope of the constant-pressure surface. The atmosphere is called barotropic, when there are no geostrophic shears or horizontal temperature gradients, and baroclinic, when there are signi®cant geostrophic shears. In the same way, atmospheric boundary layers may be classi®ed as barotropic and baroclinic PBLs. The term baroclinicity (or baroclinity) is often used as a synonym for geostrophic wind shear. It is not

94

6 Wind Distribution in the PBL

dicult to see that the presence of geostrophic shears or baroclinity in the PBL is more a rule than the exception. In strongly curved ¯ows about low- and high-pressure systems, the centripetal force alters the normal geostrophic balance between pressure gradient and Coriolis forces. Consequently, the actual wind can be signi®cantly more or less than the geostrophic wind in strongly cyclonic or anticyclonic ¯ows. The so-called `gradient wind' provides a much better approximation to the actual winds in strongly curved ¯ow systems (Hess, 1959, Chapter 12; Holton, 1992, Chapter 3). Example Problem 1 Surface isobars at a 308N latitude site during a mid-day period run east±west, indicating a horizontal pressure gradient of 3 mb per 100 km toward the south. At this time, surface isotherms are nearly perpendicular to isobars indicating a horizontal temperature gradient of 1.58C per 100 km toward the west. A temperature sounding indicated the PBL height of 2000 m and a temperature drop of 258C across the PBL from the near-surface temperature of 358C. The observed surface pressure is 1020 mb. Calculate the geostrophic winds (both magnitude and direction) at (a) the surface and (b) top of the PBL. Solution The Coriolis parameter for the site is given by f ˆ 2O sin f ˆ 2 

2p  sin 30 % 0:73  10 24  3600

4

s

1

Air density near the surface can be determined from the equation of state rˆ

P 1020  102 % 1:15 kg m ˆ RT 287:04  308:2

3

(a) In order to calculate the surface geostrophic wind from the observed pressure gradient, we use a geographical coordinate system with the x axis pointed toward the east and the y axis toward the north. In this coordinate system, @P ˆ 0; @x

@P ˆ @y

3  10

5

mb m

1

ˆ

3  10

3

Pa m

1

6.3 Friction E€ects on the Balance of Forces

95

Therefore Vg ˆ 0; Ug ˆ

1 @P 104  3  10 ˆ rf @y 1:15  0:73 % 35:74 ms

3

1

Thus, the magnitude of the surface geostrophic wind, G0 = 35.74 m s71, and its direction is from the west (2708). (b) Before calculating the geostrophic wind at the top of the PBL (h = 2000 m), we need to calculate the thermal wind components from Equation (6.4) with Dz = h = 2000 m. From the observations, we have @T ˆ @x

1:5  10

5

K m 1;

@T ˆ0 @y

The average temperature in the PBL, T = 295.7 K. Thus, DVg ˆ

hg @T % 13:63 m s 1 ; fT @x

DUg ˆ 0

Then, the geostrophic wind components at the top of the PBL can be determined as Ugh = Ug0 + DUg = 35.74 m s71 Vgh = Vg0 + DVg =713.63 m s71 Therefore, the magnitude of the geostrophic wind, Gh = [(35.74)2 + (13.63)2]1/2 % 38.25 m s71 and its direction is 2918 (NW). 6.3 Friction E€ects on the Balance of Forces Because air is a viscous ¯uid, its velocity relative to the earth's surface must vanish right at the surface. This does not happen abruptly, but the retarding in¯uence of the surface on atmospheric winds pervades the whole depth of the PBL. Consequently, the wind speed decreases gradually and the mean momen-

96

6 Wind Distribution in the PBL

tum is transferred downward in going toward the surface. Turbulence provides an ecient mechanism for the transfer of momentum from one level to another in the vertical. Also, vertical exchange of horizontal momentum results in the so-called frictional force on any ¯uid element. Because of this, the simple geostrophic balance between the pressure gradient and Coriolis forces, which may exist outside the PBL, cannot be expected to occur within the PBL. Figure 6.4 is a schematic of the balance of pressure gradient, Coriolis, and friction forces on ¯uid elements at di€erent levels in a barotropic PBL. The corresponding geostrophic and actual wind vectors are also shown with the stipulation that the Coriolis force must always be normal to V and proportional to wind speed, while the friction force must be perpendicular to the ageostrophic wind vector (V 7 G). These conditions follow from the equation of mean motion in the absence of local and advective accelerations. 2rO 6 (V 7G) = @t/@z

(6.7)

Figure 6.4 Schematic of the balance of forces on a ¯uid parcel at di€erent heights in a barotropic PBL: (a) at the surface; (b) in the surface layer; (c) in the middle of the PBL; (d) at the top of the PBL. Actual and geostrophic velocities are also indicated. [After Arya (1986).]

6.3 Friction E€ects on the Balance of Forces

97

Note that in a rotating frame of reference or in the presence of directional shear, the friction force on a ¯uid element need not be parallel and opposite to the velocity vector, as commonly depicted in many textbook schematics of the force balance in the friction layer. Figure 6.4a shows that, right at the surface where the Coriolis force disappears, the friction force must exactly balance the pressure gradient force and, hence, make a large angle with the direction of near-surface wind or surface shear. As wind speed increases with height, with little change in the wind direction through the surface layer, the balance of forces requires that the friction force decreases and rotates anticyclonically with increasing height (Figure 6.4b). The same tendency continues through the outer part of the PBL also, as wind veers and increases in magnitude (it often becomes super-geostrophic in the middle of the PBL), as shown in Figure 6.4c. Near the top of the PBL, the friction force becomes small in magnitude and highly variable in direction, in response to the changes in the direction of V 7 G, as velocity oscillates about the geostrophic equilibrium. Ignoring any such oscillations, as well as any inertial oscillations that may be present in the real atmosphere, the force balance at the top of the PBL is depicted in Figure 6.4d. The above considerations of the balance of forces suggest that winds must veer with height in a barotropic PBL. The di€erence between the wind direction at some height and that near the surface is called the veering angle, which normally increases with height in a barotropic PBL. In the presence of thermal winds (geostrophic wind shear), however, actual winds may veer or back with increasing height, in response to the veering or backing of geostrophic winds and the frictional veering. In practice, frictional veering may be estimated as the di€erence between the actual wind veering and the geostrophic veering. The angle between the directions of geostrophic and actual winds is called the cross-isobar angle (a) of the ¯ow, which normally decreases with height and vanishes at the top of the PBL (assuming that geostrophic balance occurs there). The surface cross-isobar angle (a0) which near-surface winds make with surface isobars is of considerable interest to meteorologists. It is found to depend on the surface roughness, latitude, and the surface geostrophic wind, as well as on the stability, baroclinity, and height of the PBL. For the neutral, barotropic PBL in middle and high latitudes, a0 is found to range from about 108 for relatively smooth (e.g., water, ice, and snow) surfaces to about 358 for very rough (e.g., forests and urban areas) surfaces. It increases with increasing stability and decreasing latitude, so that a wide range of a0 % 0±458 occur in the strati®ed barotropic PBL. In the presence of thermal winds (baroclinic PBL), the range of observed surface cross-isobar angles becomes wider (say, 720 to 708). Under certain conditions of strong baroclinity, a0 may even become negative, implying that a component of the near-surface winds is directed toward the surface `high' (Arya and Wyngaard, 1975; Arya, 1978).

98

6 Wind Distribution in the PBL

Example Problem 2 The following observations are for a midlatitude ( f = 1074 s71) PBL during the afternoon convective conditions: Height (m) Wind speed (m s71) Wind direction (deg.):

10 9.0 250

100 15.5 253

1000 17.0 258

1500 15.0 260

The PBL height = 1500 m. The surface geostrophic wind speed = 20.0 m s71. The surface geostrophic wind direction = 2758. (a) Calculate the average magnitudes of the geostrophic wind shear and the actual wind shear in the PBL, assuming geostrophic balance at the top of the PBL. (b) Estimate the total wind veering and the frictional veering across the PBL, as well as the cross-isobar angle of the near-surface winds. (c) Compute the pressure gradient, Coriolis, and friction forces per unit mass at the 10 m level and show the force balance at this level. Solution (a) Using the geographical coordinate system with the x-axis pointed toward the east and the y-axis toward the north, the horizontal components of geostrophic winds at the surface and the top (h = 1500 m) of the PBL are: Ug0 = G0 cos (270 7 275) = 19.92 m s71 Vg0 = G0 sin (270 7 275) = 71.74 m s71 Ugh = Gh cos (270 7 260) = 14.77 m s71 Vgh = Gh sin (270 7 260) = 2.60 m s 71 The components of the geostrophic wind shear across the PBL can, then, be calculated as DUg = 14.77 7 19.92 = 75.15 m s71 DVg = 2.60 7 (71.74) = 4.34 m s71 So that, |DG| = [(5.15)2 + (4.34)2]1/2 = 6.73 m s71 The magnitude of the geostrophic wind shear across the PBL = |DG|/h % 4.49 6 1073 s71. The average magnitude of the actual wind shear jVh j ˆ 10 h

2

s

1

6.4 Stability E€ects on the PBL Winds

99

(b) The total (actual) wind veering = 260 7 250 = 108. The geostrophic wind veering = 260 7 275 = 7158. Frictional veering = total veering 7 geostrophic veering = 10 7 (715) = 258 The cross-isobar angle of near-surface winds = 275 7 250 = 258, which is equal to the frictional veering across the PBL. (c) The various forces per unit mass or accelerations are given as follows: Type of force

Magnitude

Direction

Pressure gradient (P) Coriolis force (C) Friction force (F)

f |G| f |V| f |V 7 G|

Normal to G oriented to the left of G Normal to V oriented to the right of V Normal to (V 7 G)

In the absence of any local or advective acceleration, the above forces must balance, so that, at any height, P + C + F = 0. The magnitudes of the various forces at the 10 m level are: |P| = f |G| = 2.0 6 1073 m s72 |C| = f |V| = 0.9 6 1073 m s72 |F| = f |V 7 G| = 1.24 6 1073 m s72 Note that |V 7 G] = |V| 7 |G| but |V 7 G] = [(U 7 Ug)2 + (V 7 Vg)2]1/2 The force balance can easily be shown in a vector plot similar to Figure 6.4(b), knowing the magnitudes and directions of geostrophic and actual winds. 6.4 Stability E€ects on the PBL Winds We have discussed earlier how the diurnal cycle of heating and cooling of the surface, in conjunction with radiative, convective, and advective processes occurring within the PBL, determine the static or thermal stability of the PBL. Through its strong in¯uence on the vertical movements of air parcels, thermal stability strongly in¯uences the vertical exchange of momentum and, hence, the wind distribution in the PBL.

100

6 Wind Distribution in the PBL

On a clear day the surface warms up relative to the air above, in response to solar heating. This gives rise to a variety of convective circulations, such as nearsurface plumes, thermals (updrafts) and downdrafts, which can directly transfer momentum and heat in the vertical direction. The upward transfer of heat through the lower part of the PBL is largely responsible for convective or buoyancy-generated turbulence. The resulting vigorous mixing of momentum leads to considerable weakening and, sometimes, elimination of mean wind shears in the PBL. Thus, it is not uncommon to ®nd nearly uniform wind speed and wind direction pro®les through much of the daytime convective boundary layer (CBL). Strong wind shears are largely con®ned to the lower part of the surface layer and the shallow transition layer near the base of inversion which often caps the convective mixed layer. Changes in mean wind direction between the surface and the inversion base are typically less than 208. On a clear evening and night, on the other hand, the surface cools down in response to longwave radiation and a surface inversion begins to form and develop. Since buoyancy inhibits vertical momentum exchanges in the inversion layer, signi®cant wind speed and direction shears develop in this layer. Wind direction changes of 308 or more across the shallow stable boundary layer (SBL), which may comprise only a part of the surface inversion layer, are not uncommon. The wind speed pro®le is generally characterized by a low-level jet in which winds are often supergeostrophic. The winds in the outer part of the SBL may undergo inertial oscillations in response to the inertially oscillating geostrophic ¯ow. Internal gravity waves may also develop in such a strati®ed environment; such waves frequently appear mixed with turbulence. Thermal stability also has a strong in¯uence on the PBL height, which, in turn, a€ects the wind distribution. In the morning hours following sunrise, the PBL grows rapidly at ®rst, in response to heating from below. This growth continues throughout the day, although at a progressively decreasing rate, resulting in a 1- to 2-km-deep mixed layer by mid-afternoon. Immediately following the evening transition period, when the sensible heat ¯ux at the surface changes sign, the unstable PBL suddenly collapses and is replaced by a much shallower stable boundary layer. For a given wind speed at the top of the PBL, one would expect the average wind shear in the PBL to be an order of magnitude larger at night than that during the midday period, because of the di€erence in the PBL heights. Very close to the surface, however, wind shears are generally larger during the daytime than they are at night. In the presence of wind shear, a dynamic stability parameter such as the gradient Richardson number g @Yv @V Ri ˆ Tv @z @z

2

…6:8†

6.5 Observed Wind Pro®les

101

is considered to be a more appropriate stability parameter than the static stability parameter s = (g/Tv)(@Yv/@z) introduced earlier. Ri is dimensionless and has the same sign as s. It will be shown in Chapter 8 that the Richardson number is a better measure of the intensity of mixing (turbulence) and provides a simple criterion for the existence or nonexistence of turbulence in a stably strati®ed environment (a large positive value of Ri 4 0.25 is indicative of weak and decaying turbulence or a completely nonturbulent environment). Therefore, the vertical distribution of Ri may be used to determine the vertical extent of the boundary layer when other, more direct, measurements of the PBL height are not available. 6.5 Observed Wind Pro®les A considerable amount of wind pro®le data have been collected by meteorologists in the course of a number of large and small ®eld experiments (Garratt and Hicks, 1990; Tunick, 1999), as well as from routine upper air soundings. Observations are made using pilot balloons, rawindsondes, tethersondes, doppler radars, acoustic sounders (sodars), instrumented aircraft, and tall towers. Consequently, scientists have developed a good understanding of the e€ects of surface roughness and stability on wind distribution in the PBL and a fair understanding of the in¯uence of baroclinity on the same. More complicated e€ects of entrainment, advection, complex topography, clouds, and precipitation have received little attention from boundary layer meteorologists and remain poorly understood. The underlying philosophy in micrometeorology has been to study simple situations ®rst in order to have a better understanding of the basic phenomena and then include progressively more complicating factors. Unfortunately, diculties of measuring and computing turbulence in the atmospheric boundary layer have restricted micrometeorologists for a long time to the study of the more or less idealized (horizontally homogeneous, stationary, nonentraining, dry, etc.) PBL. Only recently have e€orts been made to study the e€ects of complicating factors which in¯uence the real-world PBL. First, we present a few selected wind pro®les for the `ideal' PBL, taken under di€erent stability conditions, in order to illustrate some qualitative e€ects of stability. Figure 6.5 shows the pibal wind and potential temperature pro®les under fairly convective conditions during day 33 of the Wangara Experiment in southern Australia. Here, U and V are the horizontal wind components in the x and y directions, respectively, with the x axis parallel to the direction of nearsurface winds and the y axis normal to the same (this choice of coordinate axes is more commonly used in micrometeorology than the geographical coordinate system, which is used in other branches of meteorology).

102

6 Wind Distribution in the PBL

Figure 6.5 Measured wind, potential temperature and speci®c humidity pro®les in the PBL under convective conditions on day 33 of the Wangara Experiment. [From Deardor€ (1978).]

Note that despite the large geostrophic shear present on this day, wind pro®les show the characteristic features of nearly uniform distributions in the convective mixed layer (the undulations in the PBL are probably caused by large eddies and organized updraft and downdraft motions whose e€ects are not smoothed out in pibal soundings), and strong gradients in the surface layer below and the transition layer above. Similar features have been observed in convective boundary layers over other homogeneous land and ocean surfaces (Figure 6.6a). The observed mean wind and virtual potential temperature pro®les under moderately unstable conditions in a trade wind marine PBL are shown in Figure 6.6b. At the time of these observations widely scattered shallow convective clouds were present whose bases coincided with the inversion base. Note that although the Yv pro®le is uniform throughout the subcloud layer, the wind pro®les show signi®cant shears in this layer. Still, the total wind veering across the unstable PBL remains small (about 88), due to the combined e€ects of mechanical and buoyant mixing, smooth ocean surface, and low latitude. The theoretician's ideal of a steady state, neutral, barotropic PBL is so rare in the atmosphere that relevant observations of the same do not exist. The wind pro®les taken under slightly unstable and slightly stable conditions during the morning and evening transition hours di€er considerably, due to the e€ects of nonstationarity and even slight stability or instability. An average of pro®les taken during completely overcast and very windy conditions over sea might be more representative of a neutral PBL (Nicholls, 1985). From observations of the nocturnal stable boundary layer (SBL), one can perhaps distinguish between the two broad stability regimes: (1) the moderately stable regime in which turbulent exchanges are more or less continuous in time and space through at least the lower half of the SBL; and (2) the very stable

6.5 Observed Wind Pro®les

103

Figure 6.6 Observed vertical pro®les of mean wind components, or wind speed and direction, potential temperature and speci®c humidity in the PBL. (a) Convective conditions overland. [After Kaimal et al. (1976).] (b) Unstable conditions over the ocean. [After Pennell and LeMone (1974).]

regime in which turbulent exchanges occur only intermittently in time and space through most of the SBL. The former can exist over land at night only during strong winds, and more likely at sea when the air is slightly warmer than the sea surface. The typical pro®les of wind, potential temperature, and Richardson number in this moderately stable boundary layer are shown in Figure 6.7. These are based on the measurements from a tall tower near Dallas, Texas. Note that the U pro®le shows a pronounced maximum. The nose of the low-level jet often coincides with the top of the SBL, since it also represents the level of maximum

104

6 Wind Distribution in the PBL

Figure 6.7 Observed vertical pro®les of mean wind components and potential temperature and the calculated Ri pro®le in the nocturnal PBL under moderately stable conditions. [From Deardor€ (1978); after Izumi and Barad (1963).]

in Ri. The Ri pro®le indicates that continuous turbulence may be expected only in the lower half of the SBL, where Ri 5 0.25. The sporadic turbulence regime is more common in the SBL over land. The typical observed pro®les of U, V, Y , and Ri in the very stable regime are shown in Figure 6.8. Note that both the wind components attain maximum values at low levels. The PBL depth based on the maximum in the wind speed pro®le is only about half of the depth of the surface inversion layer. The Ri pro®le indicates that the layer of possible continuous turbulence in which Ri 5 0.25 is very shallow indeed and in the bulk of the SBL turbulence is likely to be intermittent, patchy, and weak.

Figure 6.8 Observed wind and potential temperature pro®les under very stable (sporadic turbulence) conditions at night during the Wangara Experiment. [From Deardor€ (1978).]

6.5 Observed Wind Pro®les

105

Figure 6.9 Composite pro®les of wind speed, potential temperature and Richardson number scaled with respect to the low-level jet height at (a) O'Neill, Nebraska, during the Great Plains Experiment and (b) Hay, Australia, during the Wangara Experiment. [After Mahrt et al. Copyright # (1979) by D. Reidel Publishing Company. Reprinted by permission.]

The occurrence of a low-level jet is a common phenomenon in the SBL (Figure 6.9). The jet can become highly intensi®ed when thermal or slope winds oppose the surface geostrophic wind. Such intensi®ed low-level jets have frequently been observed over the Great Plains region of the United States. Figure 6.9 shows low-level jets in the composite wind speed pro®les obtained

106

6 Wind Distribution in the PBL

from observations during the Wangara Experiment (Clark et al., 1971) and the Great Plains Field Program (Lettau and Davidson, 1957). Other characteristic features of the stably strati®ed PBL are large directional shear (wind veering) and small PBL depth, both of which are manifestations of the e€ects of stability. The veering of wind with height is better illustrated through a representation of wind components in the form of a wind hodograph, which is the locus of the end points of velocity vectors at various heights. Figure 6.10 presents such hodographs from a 32 m tower at Plateau Station, Antarctica. These are actually averages of many observed wind pro®les grouped under eight stability classes (from the least stable class 1 to the most stable class 8) and three periods or seasons. The corresponding temperature pro®les have already been given in Figure 5.5. Note that with increasing stability, wind veering with height also increases. For extremely stable classes changes in wind direction with height can be detected even at low levels of 1 or 2 m, so that the surface layer becomes very shallow under these conditions. The close relationship between wind veering and stability or temperature lapse rate is further demonstrated by observations given in Figure 6.11. Note that wind veering increases with increasing stability (negative lapse rate); the negative values (backing of wind) under superadiabatic conditions are probably due to the in¯uence of thermal winds.

Figure 6.10 Averaged observed wind hodographs at Plateau Station, Antarctica, for (a) sunlight, (b) transitional, and (c) dark periods, grouped under di€erent stability classes 1±8. Dots indicate ends of wind vectors at heights of 0.5, 1, 2, 4, 8, 12, 16, 20, 24, and 32 m. The components shown are in the so-called geotriptic coordinate system. [After Lettau et al. (1977).]

6.6 Diurnal Variations

107

Figure 6.11 Observed relationship between wind veering (± ± ±) and lapse rate (ÐÐ) near Oklahoma City. [After Gray and Mendenhall (1973).]

6.6 Diurnal Variations Diurnal variations of wind speed and wind direction in the PBL have been inferred from observations collected at di€erent locations. The evaluation of wind pro®les in the course of a diurnal cycle may show large day-to-day variations due to changes in the synoptic weather situation and the surface energy balance. When averaged over long periods of time (order of a month or longer), however, the diurnal variations are better discerned. Figure 6.12 shows the diurnal variations of mean wind speed (averaged over a period of 1 year) observed from an instrumented 500 m tower near Oklahoma City, Oklahoma. The range of times of sunrise and sunset, as well as the observation heights, are indicated in the ®gure. Note that the near-surface wind speed increases sharply after sunrise, attains a broad maximum in early afternoon, and decreases sharply near sunset. The diurnal wave at the higher level in the surface layer is similar, but with a reduced amplitude. The increase in the strength of surface winds following the morning inversion breakup is due to more rapid and ecient transfer of momentum from aloft through the evolving unstable or convective PBL in the daytime.

108

6 Wind Distribution in the PBL

Figure 6.12 Diurnal variations of (a) mean wind speed and (b) mean wind direction, on an annual basis, for various height levels in the PBL near Oklahoma City, Oklahoma. [After Crawford and Hudson (1973).]

Above the surface layer, the diurnal wave becomes nearly 1808 out of phase and wind speed decreases sharply following sunrise, attains a minimum value around noon, and increases in the afternoon and evening hours. The amplitude of the wave increases with height and attains its maximum value somewhere in the middle of the convective PBL. Although these observations did not extend to higher levels, the amplitude of diurnal variation of wind speed is expected to decrease farther up and vanish at the maximum height of the daytime PBL. Similar diurnal patterns of wind speed at various heights in the PBL are shown in Figure 6.13, which is based on 40-day averages of pibal observed winds during the Wangara expedition at a smooth rural site in Australia. It can be inferred from the diurnal evolution of potential temperature and speci®c humidity pro®les shown in Chapter 5 and, to some extent, from the wind pro®les also, that PBL depth has a strong diurnal variation, especially during the daytime under clear skies. The unstable or convective PBL is usually capped by an inversion and the height of the inversion base zi is considered to be a measure of the PBL depth h (actually, h is 5±30% larger than zi, if one includes the interfacial transition layer in the former). The diurnal evolution of the height of the lowest inversion base on a typical day during the 1973 Minnesota experiment is shown in Figure 6.14. Also represented in the same ®gure is the diurnal variation of the surface heat ¯ux H0 whose cumulative input (integration with respect to time) to the PBL is primarily responsible for the growth of zi or h.

6.7 Applications

109

Figure 6.13 Diurnal variations of 40-day-averaged wind speeds at various heights in the PBL during the Wangara Experiment. [After Mahrt (1981).]

Figure 6.14 Typical variations of (A) the height of inversion base and (B) sensible heat ¯ux during the daytime period in the Minnesota Experiment. [After Kaimal et al. (1976).]

6.7 Applications The knowledge of wind distribution in the PBL has the following practical applications:

110

6 Wind Distribution in the PBL

. Determining the rate of dissipation of the kinetic energy in the lower atmosphere. . Determining local and regional transports of pollutants and air trajectories in the lower atmosphere. . Estimating momentum ¯ux or shear stress through the PBL and the surface drag. . Estimating wind energy potential and designing wind power-generating systems. . Designing tall buildings, towers, and bridges for wind loads. . Designing wind shelters and other protective measures. Problems and Exercises 1. Derive the thermal wind equations from the geostrophic wind relations, the hydrostatic equation, and the equation of state for the lower atmosphere, and indicate when and why the terms involving the vertical temperature gradient may be neglected. 2. Show that the net horizontal friction force per unit volume of a ¯uid element in the PBL is @t/@z and that it need not be opposite to the velocity vector V. How is the friction force related to the ageostrophic wind vector? 3. The following measurements of mean winds and temperatures were made from the 200 m mast at Cabauw in The Netherlands on a September night: Height (m)

Wind speed (m s71)

Wind direction (deg)

Temperature (8C)

5 10 20 40 80 120 160 200

2.2 2.7 3.5 5.4 8.2 9.1 8.7 7.8

194 194 202 217 234 240 246 250

8.58 8.81 9.08 10.15 12.19 12.96 12.86 12.63

Geostrophic wind speed at the surface = 5.94 m s71 Geostrophic wind direction at the surface = 2638 Horizontal temperature gradient toward east = 72.69 6 1075 K m71 Horizontal temperature gradient toward north = 8.85 6 1076 K m71

Problems and Exercises

111

Boundary layer depth = 100 m Surface pressure = 1000 mbar (a) Plot on a graph the wind speed, wind direction, and potential temperature pro®les. (b) Plot the wind hodograph, using the geographical coordinate system, and indicate geostrophic wind vectors at the surface and the top of the PBL. Compare the actual wind with the geostrophic wind at 100 m. (c) Calculate and plot the wind component pro®les, taking the x axis along the near-surface (5 m) wind. 4. (a) Using the wind component and potential temperature pro®les of the previous problem, determine the magnitudes of wind shear, potential temperature gradient, and Richardson number at the heights of 7.5, 15, 30, 60, 100, 140, and 180 m. (b) Plot Ri as a function of height and indicate the probable top of the PBL based on the maximum in the Ri pro®le. 5. The following observations are for a midlatitude ( f = 1074 s71) PBL during the afternoon convective conditions: Heighta (m)

Wind speedb (m s71)

Wind directionc (deg)

1 32 61 610 1220

7.7 11.0 11.4 11.7 12.1

311 315 315 321 331

a

PBL height = 1220 m. Surface geostrophic wind speed = 10.0 m s71. c Surface geostrophic wind direction = 2958. b

(a) Calculate the average magnitudes of geostrophic wind shear and the actual wind shear in the PBL, assuming geostrophic balance at the top of the PBL. (b) Compute the total wind veering and frictional veering across the PBL, as well as the cross-isobar angle of the near-surface winds. (c) Compute the pressure gradient, Coriolis, and friction forces per unit volume (r = 1.2 kg m73) at the 610 m level and vectorially represent (to scale) the force balance at this level.

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6 Wind Distribution in the PBL

6. Calculate the magnitudes and directions of pressure gradient, Coriolis, and friction forces per unit mass of a ¯uid element at the top of a nocturnal surface layer where the actual winds are 7.07 m s71 from the west, the geostrophic wind speed is 10 m s71, and the cross-isobar angle is 458 (take f = 1074 s71).

Chapter 7

An Introduction to Viscous Flows

7.1 Inviscid and Viscous Flows In the preceding two chapters, our treatment of temperature, humidity, and wind distributions in the PBL was largely empirical and based on rather limited observations. A better understanding of vertical pro®les of wind, temperature, and humidity and their relationships to the important exchanges of momentum, heat, and moisture through the PBL has been gained through applications of mathematical, statistical, and semiempirical theories. A brief description of some of the fundamentals of viscous ¯ows and turbulence and the associated theory will be given in this and the following chapters. For theoretical treatments, ¯uid ¯ows are commonly divided into two broad categories, namely, inviscid and viscous ¯ows. In an inviscid or ideal ¯uid the e€ects of viscosity are completely ignored, i.e., the ¯uid is assumed to have no viscosity, and the ¯ow is considered to be nonturbulent. Inviscid ¯ows are smooth and orderly, and the adjacent ¯uid layers can easily slip past each other or against solid surfaces without any friction or drag. Consequently, there is no mixing and no transfer of momentum, heat, and mass across the moving layers. Such properties can only be transported along the streamlines through advection. The inviscid ¯ow theory obviously results in some serious dilemmas and inconsistencies when applied anywhere close to solid surfaces or density interfaces. Such dilemmas can be resolved only by recognizing the presence of boundary layers or interfacial mixing layers in which the e€ects of viscosity or turbulence cannot be ignored. Far away from the boundaries and density interfaces, however, the ¯uid viscosity can be ignored and the inviscid or ideal ¯ow model provides a very good approximation to many real ¯uid ¯ows encountered in geophysical and engineering applications. Extensive applications of this are given in books on hydrodynamics and geophysical ¯uid dynamics (Lamb, 1932; Pedlosky, 1979).

113

114

7 An Introduction to Viscous Flows

7.1.1 Viscosity and its e€ects Fluid viscosity is a molecular property which is a measure of the internal resistance of the ¯uid to deformation. All real ¯uids, whether liquids or gases, have ®nite viscosities associated with them. An important manifestation of the e€ect of viscosity is that ¯uid particles adhere to a solid surface as they come in contact with the latter and consequently there is no relative motion between the ¯uid and the solid surface. If the surface is at rest, the ¯uid motion right at the surface must also vanish. This is called the no-slip boundary condition, which is also applicable at the interface of two ¯uids with widely di€erent densities (e.g., air and water). Within the ¯uid ¯ow, viscosity is responsible for the frictional resistance between adjacent ¯uid layers. The resistance force per unit area is called the shearing stress, because it is associated with the shearing motion (variation of velocity) between the layers. A simple demonstration of this is provided by the smooth, streamlined, laminar ¯ow between two large parallel planes, one ®xed and the other moving at a slow constant speed, Uh, which are separated by a small distance h (see Figure 7.1a). At suciently small values of Uh and h, laminar ¯ow can be maintained and the velocity within the ¯uid varies linearly from zero at the ®xed plane to Uh at the moving plane, so that the velocity gradient @u/@z = Uh/h, everywhere in the ¯ow. From observations in such a ¯ow, Newton found the relationship that shearing stress is proportional to the rate of strain or velocity gradient (¯uids following a linear relationship between the stress and the rate of strain are known as Newtonian ¯uids), that is, t = m(@u/@z)

(7.1)

Where the coecient of proportionality m is called the dynamic viscosity of the ¯uid. In ¯uid ¯ow problems, it is more convenient to use the kinematic viscosity n : m/r, which has dimensions of L2 T71. In particular, for the gaseous ¯uids, such as air, a more rigorous theoretical derivation of the relationship between the shearing stress and velocity gradient can be obtained from the kinetic theory (Sutton, 1953), which clearly shows viscosity to be a molecular property which depends on the temperature and pressure in the ¯uid (this dependence is, however, weak and is usually ignored in most atmospheric applications). Equation (7.1) is strictly valid only for a unidirectional ¯ow. In most real ¯ows, in general, spatial variations of velocity in di€erent directions give rise to shear stresses in di€erent directions. More general constitutive relations between the components of shearing stress and velocity gradient at any point in the ¯uid are (Kundu, 1990, Chapter 4)

7.1 Inviscid and Viscous Flows

115

Figure 7.1 Schematics of velocity and shear stress pro®les in laminar plane-parallel ¯ows: (a) Couette ¯ow; (b) channel ¯ow; (c) free-surface gravity ¯ow.

txy = tyx = m(@u/@y + @v/@x) txz = tzx = m(@u/@z + @w/@x) tyz = tzy = m(@v/@z + @w/@y)

(7.2)

in which the ®rst member of the subscript to t denotes the direction normal to the plane of the shearing stress and the second member denotes the direction of the stress. Equation (7.2) implies that in Newtonian ¯uids, shear stresses are proportional to the applied strain rates or deformations (represented by

116

7 An Introduction to Viscous Flows

quantities in parentheses). Here, stresses and deformations are all instantaneous quantities at a point in the ¯ow. Another important e€ect of viscosity is the dissipation of kinetic energy of the ¯uid motion, which is constantly converted into heat. Therefore, in order to maintain the motion, the energy has to be continuously supplied externally, or converted from potential energy, which exists in the form of pressure and density gradients in the ¯uid. Although the above-mentioned e€ects of viscosity are felt, in varying degrees, in all real ¯uid ¯ows, at all times, they are found to be particularly signi®cant only in certain regions and in certain types of ¯ows. Such ¯ows are called viscous ¯ows, as opposed to the inviscid ¯ows mentioned earlier. Examples of such ¯ows are boundary layers, mixing layers, jets, plumes, and wakes, which are dealt with in many books on ¯uid mechanics (Batchelor, 1970; Townsend, 1976; Kundu, 1990). 7.2 Laminar and Turbulent Flows All viscous ¯ows can broadly be classi®ed as laminar and turbulent ¯ows, although an intermediate category of transition between the two has also been recognized. A laminar ¯ow is characterized by smooth, orderly, and slow motion in which adjacent layers (laminae) of ¯uid slide past each other with very little mixing and transfer (only at the molecular scale) of properties across the layers. The main di€erence between the laminar ¯ows and the inviscid ¯ows introduced earlier is the importance of viscous and other molecular transfers of momentum, heat, and mass in the former. In laminar ¯ows, the ¯ow ®eld and the associated temperature and concentration ®elds are regular and predictable and vary only gradually in space and time. In sharp contrast to laminar and inviscid ¯ows, turbulent ¯ows are highly irregular, almost random, three-dimensional, highly rotational, dissipative, and very di€usive (mixing) motions. In these, all the ¯ow and scalar properties exhibit highly irregular variations (¯uctuations) in both time and space, with a wide range of temporal and spatial scales. For example, turbulent ¯uctuations of velocity in the atmospheric boundary layer typically occur over time scales ranging from 1073 to 104 s and the corresponding spatial scales from 1073 to 104 m ± more than a millionfold range of scales. Due to their nearly random nature (there is some order, persistence, and correlation between ¯uctuations in time and space), turbulent motions cannot be predicted or calculated exactly as functions of time and space; one usually deals with their average statistical properties. Although there are many examples and applications of laminar ¯ows in industry, in the laboratory, and in biological systems, their occurrence in

7.3 Navier±Stokes Equations of Motion

117

natural environments, particularly the atmosphere, is rare and con®ned to the so-called viscous sublayers over smooth surfaces (e.g., ice, mud ¯ats, relatively undisturbed water, and tree leaves). Most ¯uid ¯ows encountered in nature and engineering applications are turbulent. In particular, the various smallscale motions in the lower atmosphere are turbulent. The turbulent mixing layer may vary in depth from a few tens of meters in clear, calm, nocturnal cooling conditions to several kilometers in highly disturbed (stormy) weather conditions involving deep penetrative convection. In the upper troposphere and stratosphere also, extensive regions of clear air turbulence are found to occur. Some meteorologists consider all atmospheric motions right up to the scale of general circulation as turbulent. But, then, one has to distinguish between the small-scale three-dimensional turbulence of the type we encounter in micrometeorology, and the large-scale `two-dimensional turbulence.' There are fundamental di€erences in some of their properties and mechanisms of energy transfers up or down the scales. We will only be concerned with the former here. Turbulent, too, are the ¯ows in upper oceans (e.g., oceanic mixed layer), lakes, rivers, and channels; practically all ¯ows in liquid and gas pipelines; in boundary layers over moving aircraft, missiles, ships, and boats; in wakes of structures, propeller blades, turbine blades, projectiles, bullets, and rockets; in jets of exhaust gases and liquids; and in chimney and smokestack plumes. Thus turbulence is literally all around us, particularly in the form of the atmospheric boundary layer, from which we can escape only brie¯y. The next and the following chapters will be devoted to fundamentals of turbulence and their application to micrometeorology. But ®rst we discuss the basic equations for viscous motion, which are valid for both laminar and turbulent ¯ows. 7.3 Navier±Stokes Equations of Motion Mathematical treatments of ¯uid ¯ows, including those in the atmosphere, are almost always based on the equations of motion, which are mathematical expressions of the fundamental laws of the conservation of mass, momentum, and energy. For example, considerations of mass conservation in an elemental volume of ¯uid leads to the equation of continuity, which for an incompressible ¯uid, in a Cartesian coordinate system, is given by @u/@x + @v/@y + @w/@z = 0

(7.3)

The continuity equation imposes an important constraint on the ¯uid motion such that the divergence of velocity must be zero at all times and at every point

118

7 An Introduction to Viscous Flows

in the ¯ow. The assumption of incompressible ¯uid or ¯ow is quite well justi®ed in micrometeorological applications. The application of Newton's second law of motion, or consideration of momentum conservation in an elemental volume of ¯uid, leads to the so-called Navier±Stokes equations, which, in a Cartesian frame of reference tied to the surface of the rotating earth with x and y axes in the horizontal and z axis in the vertical, are Du Dt

fv ˆ

1 @p ‡ nH2 u r @x

Dv ‡ fu ˆ Dt

1 @p ‡ nH2 v r @y

Dw ‡gˆ Dt

1 @p ‡ nH2 w r @z

…7:4†

Where the total derivative and Laplacian operator are de®ned as D/Dt : @/@t + u(@/@x) + v(@/@y) + w(@/@z)

(7.5)

H2 : @ 2/@x2 + @ 2/@y2 + @ 2/@z2

(7.6)

The derivation of the above equations of motion is outside the scope of this text and can be found elsewhere (Haltiner and Martin, 1957; Dutton, 1976; Kundu, 1990). The physical interpretation and signi®cance of each term can be given as follows. Terms in Equation (7.4) represent accelerations or forces per unit mass of the ¯uid element in the x, y, and z direction, respectively. On the left-hand sides, the ®rst terms are called the inertia terms, because they represent the inertia forces arising due to local and advective accelerations on a ¯uid element. The second terms in the equations of horizontal motion represent the Coriolis accelerations or forces which apparently act on the ¯uid element due to the earth's rotation. The rotational term is insigni®cant in the equation of vertical motion and has been omitted. Instead, the gravitational acceleration term appears in the equation for vertical motion. On the right-hand sides of Equation (7.4), the ®rst terms represent the pressure gradient forces and the second terms are viscous or friction forces on the ¯uid element. When ¯uid viscosity or friction terms can be ignored, the equations of motion reduce to the so-called Euler's equations for an inviscid or ideal ¯uid. The

7.4 Laminar Plane-Parallel Flows

119

solutions of these equations for a variety of density or temperature strati®cations and initial and boundary conditions now form a rich body of literature in classical hydrodynamics (Lamb, 1932) and geophysical ¯uid dynamics (Pedlosky, 1979). Euler's equations of motion are considered to be adequate for describing the behavior of atmosphere and oceans outside of any boundary, interfacial, and mixing layers in which the ¯ows are likely to be turbulent. For a mathematical description of viscous ¯ows, one has to use the complete set of Navier±Stokes equations, which are nonlinear partial di€erential equations of second order and are extremely dicult (often impossible) to solve. The combination of nonlinear inertia terms and viscous terms is responsible for the extreme diculty and, in many cases, the intractability of solutions. Analytical solutions have been found only for limited cases of very low-speed laminar or creeping ¯ows when nonlinear terms are absent or can be simpli®ed. Some of these `exact' solutions are given in the following sections in order to introduce the reader to certain classical ¯uid ¯ows. 7.4 Laminar Plane-Parallel Flows The simplest viscous ¯ows are the steady, one-dimensional, laminar ¯ows between two in®nite parallel planes. Since velocity varies only normal to the planes and there is no motion across the planes, the inertia terms in the Navier± Stokes equations are zero. For small-scale laboratory ¯ows, the earth's rotational e€ects (Coriolis terms) can also be ignored. Furthermore, choosing the x axis in the direction of ¯ow and the z axis normal to the plane boundaries, the equations of motion reduce to d 2u/dz2 = (1/m)(@p'/@x)

(7.7)

which can easily be solved for di€erent ¯ow situations (boundary conditions). Here, p' is the modi®ed or excess pressure obtained after subtracting the hydrostatic part from the actual pressure. For a ¯uid of uniform density, introduction of modi®ed pressure eliminates the in¯uence of gravity, so that the gravity term can be dropped from the equations of motion. The procedure of subtracting the hydrostatic part from the actual pressure is also widely used for strati®ed geophysical ¯ows. For these, the pressure term in the equation of vertical motion contains the deviation of the pressure from that of the reference medium at rest which is assumed to be hydrostatic. The above procedure should not be used for liquid ¯ows with a free surface, which are essentially forced by gravity (e.g., water ¯ow down a sloping surface).

120

7 An Introduction to Viscous Flows

7.4.1 Plane-Couette ¯ow Consider the laminar ¯ow between two parallel boundaries, one ®xed and the other moving in the x direction at a constant velocity of Uh, separated by a small distance h, as depicted in Figure 7.1a. The relevant no-slip boundary conditions are u ˆ 0;

at z ˆ 0

u ˆ Uh ;

at z ˆ h

…7:8†

The solution of Equation (7.7) satisfying the above boundary conditions is given by u ˆ Uh

z h

1 @p0 z…h 2m @x

z†

…7:9a†

or, in the dimensionless form, u z ˆ Uh h

h2 @p0 z  1 2mUh @x h

z h

…7:9b†

Note that the velocity pro®le in this so-called plane-Couette ¯ow is a combination of linear and parabolic pro®les. For the special case of zero pressure gradient (@p'/@x = 0), the velocity pro®le becomes linear, i.e., u/Uh = z/h

(7.10)

and the shear stress is equal to the surface shear stress t0 = mUh/h everywhere in the ¯ow. This ¯ow is entirely forced by the moving surface, or the relative motion between the two parallel surfaces, and is depicted in Figure 7.1a.

7.4.2 Plane-Poiseuille or channel ¯ow When both of the parallel bounding surfaces are ®xed, the appropriate boundary conditions for the unidirectional ¯ow between them are u = 0, u = 0,

at z = 0 at z = h

(7.11)

7.4 Laminar Plane-Parallel Flows

121

Then, the solution of Equation (7.7) yields a parabolic velocity pro®le uˆ

1 @p0 z…h 2m @x

z†

…7:12†

which implies a linear shear stress pro®le tˆm

@u ˆ @z

1 @p0 …h 2 @x

2z†

…7:13†

with the maximum value at the surface t0 =7(h/2)(@p'/@x). Note that in order to have a steady ¯ow through the channel, there must be a constant negative pressure gradient in the direction of the ¯ow. The velocity and shear stress pro®les in a plane channel ¯ow are schematically shown in Figure 7.1b. The solution of simpli®ed equations of motion in a cylindrical coordinate system yields similar velocity and stress pro®les in a circular pipe (Hagen± Poiseuille) ¯ow.

7.4.3 Gravity ¯ow down an inclined plane Here we consider a unidirectional liquid ¯ow with a free surface or other gravity ¯ows over uniformly sloping surfaces. The relevant equation of motion with the x axis in the direction of ¯ow (down the slope b) and the z axis normal to the surface (Figure 7.1c) is d 2u/dz2 = 7(g/n) sin b

(7.14)

in which we have included the gravity force term but have ignored the pressure gradient. The boundary conditions are u ˆ 0; at z ˆ 0 du=dz ˆ 0; at z ˆ h

…7:15†

The upper boundary condition implies zero shear stress at the free surface, or at z = h, where h represents the constant thickness of the frictional shear layer. The solution of Equation (7.14) satisfying the above boundary conditions is given by uˆ

g sin b z…2h 2n

z†

…7:16†

122

7 An Introduction to Viscous Flows

which is again a parabolic pro®le with the maximum velocity at z = h uh ˆ

gh2 sin b 2n

…7:17†

The shear stress pro®le is linear with the maximum value at the surface t0 = rgh sin b

(7.18)

These pro®les are schematically shown in Figure 7.1c. Note that both the maximum velocity and the surface shear stress are proportional to the sine of the slope angle. Example Problem 1 Consider fully-developed laminar ¯ow through a circular tube of diameter d. Using the simpli®ed forms of the equations of motion in cylindrical coordinates (r, y, x), with the x axis coinciding with the axis of the tube and r representing the radial distance from the center of the tube, obtain expressions for the velocity and shear stress distributions across the tube and compare them with those for the plane channel ¯ow. Solution Due to symmetry, all the terms in the equations of motion in r and y directions are identically zero, because there is no motion along those directions. This implies that @p/@r = 0 and @p/@y = 0. The x momentum equation reduces to 0ˆ or,

  @p0 m d du ‡ r @x r dr dr

  1d du 1 @p0 r ˆ r dr dr m @x

Since the left-hand side of the above equation can be a function only of r, and the right-hand side can be a function only of x, it follows that both terms must be constant. Therefore, the pressure gradient along the direction of ¯ow must be constant and pressure falls linearly along the length of the tube. Multiplying the above equation by r and integrating with respect to r, we get r

du r2 @p0 ‡A ˆ dr 2m @x

7.4 Laminar Plane-Parallel Flows

123

or, du r @p0 A ‡ ˆ dr 2m @x r in which the integration constant A = 0, because du/dr = 0 at r = 0 (velocity is maximum at the center of the tube). Integrating the above equation again gives uˆ

r2 @p0 ‡B 4m @x

where B is another integration constant, which can be evaluated from the no-slip boundary condition at the tube surface (r = d/2, u = 0) as B = 7(d 2/16m)@p'/@x. Thus, the velocity distribution in the tube is given by uˆ

1 @p0 2 …d 16m @x

4r2 †

The corresponding shear stress distribution is given by tˆm

@u r @p0 ˆ @r 2 @x

with the maximum value at the tube surface t0 = 0.25d @p'/@x. Note that the velocity pro®le across the tube cross-section is parabolic and the shear stress pro®le is linear, similar to those for the plane channel ¯ow given by Equations (7.12) and (7.13). The volumetric ¯ow rate is given by Z Qˆ

d=2 0

2prudr ˆ

p @p0 8m @x

Z

d=2

0

r…d 2

4r2 †dr

or, Qˆ

d 4 @p0 128m @x

The average velocity through the tube can be obtained by dividing the above volume ¯ow rate by the cross-sectional area of the tube, i.e., Vˆ

d 2 @p0 32m @x

124

7 An Introduction to Viscous Flows

7.5 Laminar Ekman Layers In this section, we consider the laminar Ekman boundary layers on rotating surfaces, particularly those of the atmosphere and oceans, aside from the conditions of their existence.

7.5.1 Ekman layer below the sea surface First, we examine the problem of drift currents set up at or just below the sea surface, in response to a steady wind stress forcing. For the sake of simplicity, surface waves are being ignored and so are the horizontal pressure and density gradients in water. Then, the equations of horizontal motion [Equation (7.4)] reduce to fv ˆ n

d 2u ; dz2

fu ˆ n

d 2v dz2

…7:19†

Assuming the x axis is in the direction of the applied surface stress t0 and the z axis is pointed vertically upward, the boundary conditions to be satis®ed are m

du @v ˆ t0 ; m ˆ 0; at z ˆ 0 dz @z u ! 0; v ! 0; as z ! 1

…7:20†

The two equations can be combined in a single equation for the complex current pc = u + iv, by multiplying the second part of Equation (7.19) by iˆ 1 and adding it to the ®rst. The resulting equation d2 c=dz2

i… f=n†c ˆ 0

…7:21†

has the solution satisfying the boundary condition at in®nity c ˆ u ‡ iv ˆ A exp‰a…1 ‡ i†zŠ

…7:22†

where a = ( f/2n)1/2 and A is a complex constant determined from the surface boundary condition to be A ˆ …t0 =2am†…1

i†

…7:23†

7.5 Laminar Ekman Layers

125

Substituting from Equation (7.23) into Equation (7.22) and separating into real and imaginary parts, one obtains p u ˆ …t0 = 2am†eaz cos…az p v ˆ …t0 = 2am†eaz sin…az

p=4† p=4†

…7:24†

Note pthat according to the above solution, the current has a maximum speed of t0/ 2am at the surface and is directed 458 in a clockwise sense from the direction of applied stress. With increasing depth (negative z) the current speed decreases exponentially and the current direction rotates in a clockwise sense. A common method of representing the current speed and direction as a function of depth below the surface is to plot a velocity hodograph as in Figure 7.2a. Note that the velocity hodograph represented by Equation (7.24) is a spiral; it is known as the Ekman spiral, after the Swedish oceanographer V.W. Ekman who ®rst derived the above solution. Although the induced current theoretically disappears only at an in®nite depth, in practice, the in¯uence of surface stress forcing becomes insigni®cant at a depth of the order of a71 = (2n/f )1/2. Conventionally, the Ekman layer depth hE is de®ned as the depth where the current direction becomes exactly opposite to the surface current direction. According to Equation (7.24), this happens at z =7pa71, so that hE = p(2n/f )1/2; at this depth the magnitude of the current has fallen to a small fraction (e7p % 0.04) of its surface value.

Figure 7.2 Velocity hodographs in laminar Ekman layers: (a) below free surface where tangential stress is applied; (b) above a rigid surface where a constant pressure gradient is applied. [From Batchelor (1970).]

126

7 An Introduction to Viscous Flows

Another parameter of interest is the net transport of water across vertical planes due to induced currents in the Ekman layer, which is given by the integral Z

1 0

…u ‡ iv†dz ˆ

i…t0 =rf †

…7:25†

Thus, the net transport is to the right and normal to the direction of applied stress; it is proportional to the stress, but independent of ¯uid viscosity. In fact, the above result also follows directly from the integration of the equations of motion (7.19) after expressing the viscous terms in the form of the vertical gradients of stresses, viz., dtzx/dz and dtzy/dz.

7.5.2 Ekman layer at a rigid surface Another example of an Ekman layer is that due to a uniform pressure gradient in the atmosphere near the surface or in the ocean near the bottom boundary. For the sake of simplicity, again, the surface is assumed to be ¯at and uniform and the ¯ow is considered laminar. In the absence of inertia terms, the equations of motion to be solved are fv ˆ

1 @p d 2u ‡n 2 r @x dz

fu ˆ

1 @p d 2v ‡n 2 r @y dz

…7:26†

Expressing pressure gradients in terms of geostrophic velocity components using Equation (6.1), Equation (7.26) can be written as f …v f…u

Vg † ˆ n…d 2 =dz2 †…u 2

2

Ug † ˆ n…d =dz †…v

Ug † Vg †

…7:27†

in which Ug and Vg have been taken as height-independent parameters, ignoring any small variations of these over the small depth of the Ekman layer. The appropriate boundary conditions, assuming geostrophic balance outside the Ekman layer, are u = 0 v = 0, at z = 0 u ? Ug, v ? Vg, as z ? ?

(7.28)

7.5 Laminar Ekman Layers

127

The previous set of equations [Equation (7.27)] is similar to Equation (7.19), and the solution, following the earlier solution, is u v

Ug ˆ Vg ˆ e

e az

az

‰Ug cos…az† ‡ Vg sin…az†Š

‰Ug sin…az†

Vg cos…az†Š

…7:29†

This is a general solution which is independent of the orientation of the horizontal coordinate axes. A more speci®c and simpler solution is usually given by taking the x axis to be oriented with the geostrophic wind vector, so that Ug = G and Vg = 0 in this geostrophic coordinate system and Equation (7.29) can be written as u ˆ G‰1 v ˆ Ge

e az

az

cos…az†Š

sin…az†

…7:30†

The normalized wind hodograph (Ekman spiral), according to Equation (7.30), is shown in Figure 7.2b. It shows that, in the northern hemisphere, the wind vector rotates in a clockwise sense as the height increases, and that the angle between the surface wind or stress and the geostrophic wind is 458 (this is also the cross-isobar angle of the ¯ow near the surface), irrespective of stability. Figure 7.3a shows the pro®les of the normalized velocity components u/G and v/G as functions of the normalized height az. Both the components increase approximately linearly with height above the surface, attain their maxima at di€erent heights and then oscillate around their geostrophic values with decreasing amplitude of oscillation as height increases, re¯ecting the in¯uence of the e7az factor. It should be recognized that although the spiral shape of the velocity hodograph remains invariant with the rotation of horizontal (x±y) coordinate axes, the velocity component pro®les and their expressions depend on the choice of the coordinate axes. For example, Equations (7.30) are the particular expressions of velocity components in the geostrophic coordinate system. In the surface-layer coordinate system with the x axis along the near-surface wind, which ispmore frequently  p used in micrometeorology, it is found that Ug ˆ G= 2, Vg ˆ G= 2, and the horizontal velocity components from Equation (7.29) are given as G u ˆ p 2 vˆ

G p e 2

az

G G p ‡ p e 2 2

‰cos…az†

sin…az†Š …7:31†

az

‰cos…az† ‡ sin…az†Š

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7 An Introduction to Viscous Flows

Figure 7.3 Horizontal velocity component pro®les in the laminar Ekman layer in (a) the geostrophic coordinate system, and (b) the surface-layer coordinate system. Velocity components are normalized by geostrophic wind speed G, while az is the normalized height.

These component pro®les (normalized by G) are represented in Figure 7.3b as functions of az and can be contrasted with those of Figure 7.3a based on the geostrophic coordinate system. The Ekman layer depth again is given as hE = p(2n/f )1/2, which in middle latitudes ( f % 1074 s71) is only about 1.7 m (the depth of the wind-induced

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129

oceanic Ekman spiral is about 0.5 m). There is no observational evidence for the occurrence of such shallow laminar Ekman layers in the atmosphere or oceans. Therefore, the above solutions would be of academic interest only, unless n is replaced by a much higher e€ective (eddy) viscosity K to match the theoretical Ekman layer depth with that observed under a given set of conditions. Ekman (1905) originally proposed this method or procedure for estimating the `virtual viscosity' from the observed depth, h, of frictional in¯uence as K = fh2/2p2. Even then, certain features of the above solution remain inconsistent with observations. For example, the cross-isobar angle of the atmospheric ¯ow near the surface is highly variable, as discussed in Chapter 6, and the theoretical value of a0 = 458 is found more as an exception rather than the rule. Another inconsistent feature of the Ekman solution is an almost linear velocity pro®le near the surface, while observations indicate approximately logarithmic or loglinear velocity pro®les in the surface layer. Slightly modi®ed solutions with an appropriate choice of e€ective viscosity and lower boundary conditions (in the form of speci®ed wind speed or wind direction) at the top of the surface layer have been given in the literature and largely remove the above-mentioned limitations and inconsistencies of the original Ekman solution and show better correspondence with observed wind pro®les. Example Problem 2 For the barotropic atmospheric boundary layer, the equations for the mean horizontal motion are similar to Equations (7.27) with n replaced by an apparent or e€ective viscosity K. Obtain a solution to the same in a surfacelayer coordinate system with the x axis oriented along the near-surface wind, using a lower boundary condition based on the observed near-surface wind Us at z = hs. Then, calculate the component velocity pro®les and the boundary layer height for the following parametric values: G = 10 m s71; Us = 7 m s71; K = 2 m2 s71; f = 1074 s71

hs = 10 m;

Solution The equations of mean motion for the PBL over a homogeneous surface are: rf…V rf …U

dtzx dz dtzy Ug † ˆ dz Vg † ˆ

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7 An Introduction to Viscous Flows

Expressing tzx = rK dU/dz and tzy = rK dV/dz, in analogy with their expressions (7.2) for laminar ¯ows, and assuming K to be independent of height (a rather gross assumption), one can write the above equations in the form d 2U f ‡ …V dz2 K

Vg † ˆ 0

d 2V dz2

Ug † ˆ 0

f …U K

For the barotropic PBL in which Ug and Vg are independent of height, the above two equations can be combined into one equation for the complex ageostrophic wind   d 2 Wa if Wa ˆ 0 2 dz K where, Wa ˆ …U

Ug † ‡ i…V

Vg †

The boundary conditions to be satis®ed by the solution to the above equation are: at z = hs, U = Us, and V = 0 as z ? ?, U ? Ug and V ? Vg For convenience, we can consider a new height variable z' = z 7 hs and express the ordinary di€erential equation and boundary conditions as follows: d 2 Wa dz02

if Wa ˆ 0 K

at z' = 0, Wa = Us 7Ug 7 iVg as z' ? ?, Wa ? 0 The general solution to the above di€erential equation is Wa ˆ A exp‰ …1 ‡ i†az0 Š ‡ B exp‰…1 ‡ i†az0 Š in which the constants A and B are determined from the boundary conditions as A ˆ Us

Ug

iVg ; B ˆ 0

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131

Substituting these in the above solution and decomposing it into real and imaginary parts, we obtain e

az0

‰…Ug

Us † cos…az0 † ‡ Vg sin…az0 †Š

V ˆ Vg ‡ e

az0

‰…Ug

Us † sin…az0 †

U ˆ Ug

Vg cos…az0 †Š

An important physical condition to be satis®ed by the velocity distribution in the PBL is that the shear stress vector must be parallel to the wind vector near the surface. This requires that @V ˆ0 @z0

at z0 ˆ 0; tzy ˆ 0; or

This condition yields the following relationship between the geostrophic wind components and the actual near-surface wind: Ug + Vg = Us combining it with the usual relationship U2g ‡ V2g ˆ G2 the geostrophic wind components can be determined as Ug ˆ 12‰Us ‡ …2G2

U2s †1=2 Š

…2G2

U2s †1=2 Š

Vg ˆ 12‰Us

after ignoring the other unphysical roots of the quadratic equations for Ug and Vg (note that, in the northern hemisphere, Ug 47 Vg 4 0). Using the given parameter values, we can obtain from the above expressions Ug = 9.64 m s71 and Vg =72.64 m s71 which imply that the cross-isobar angle near the surface (at z = hs, or z' = 0) ao = tan71(7Vg/Ug) % 15.48 is much smaller than 458 given by the original Ekman solution. This is a consequence of the di€erent lower boundary condition used here.

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7 An Introduction to Viscous Flows

The Ekman boundary layer height is given as  1=2 2K % 628 m hˆp f Then, using the already derived expressions for U and V, it is easy to calculate and plot the pro®les of U and V as functions of z from 10 m to h or even a higher level, if desired.

7.6 Developing Laminar Boundary Layers In the simple cases of plane-parallel ¯ows discussed in Sections 7.4 and 7.5, inertial accelerations or forces are zero and there is a balance between the pressure gradient and friction forces, or between pressure gradient, Coriolis, and friction forces. As a result, thickness of the a€ected ¯uid layer does not change in the ¯ow direction or x±y plane. In developing viscous ¯ows, such as boundary layers, wakes, and jets, the thickness of the layer in which viscous or friction e€ects are important changes in the direction of ¯ow, and inertial forces are as important, if not more, as the viscous forces. The ratio of the two de®nes the Reynolds number Re = UL/n, where U and L are the characteristic velocity and length scales (there may be more than one length scale characterizing the ¯ow). This ratio is a very important characteristic of any viscous ¯ow and indicates the relative importance of inertial forces as compared to viscous forces. Atmospheric boundary layers developing over most natural surfaces are characterized by very large (106±109) Reynolds numbers. Boundary layers encountered in engineering practice also have fairly large Reynolds numbers (Re = 103±106, based on the boundary layer thickness as the length scale and the ambient velocity just outside the boundary layer as the velocity scale). One would expect that in such large Reynolds number ¯ows the nonlinear inertia terms in the equations of motion will be far greater in magnitude than the viscous terms, at least in a gross sense. Still, it turns out that the viscous e€ects cannot be ignored in order to satisfy the no-slip boundary condition and to provide a smooth transition, through the boundary layer, from zero velocity at the surface to ®nite ambient velocity outside the boundary layer. Recognizing this, Prandtl (1905) ®rst proposed an important boundary layer hypothesis which states that, under rather broad conditions, viscosity e€ects are signi®cant in layers adjoining solid boundaries and in certain other layers (e.g., mixing layers and jets), the thicknesses of which approach zero as the Reynolds number of the ¯ow approaches in®nity, and are small outside these layers. This

7.6 Developing Laminar Boundary Layers

133

hypothesis has been applied to a variety of ¯ow ®elds and is supported by many observations. The thickness of a boundary or mixing layer should be looked at in relation to the distance over which it develops. This explains the existence of relatively thick boundary layers in the atmosphere, in spite of very large Reynolds numbers characterizing the same. The boundary layer hypothesis, for the ®rst time, explained most of the dilemmas of inviscid ¯ow theory, and clearly de®ned regions of the ¯ow where such a theory may not be applicable and other regions where it would provide close simulations of real ¯uid ¯ows. It also provided a practically useful de®nition of the boundary layer as the layer in which the ¯uid velocity makes a transition from that of the boundary (zero velocity, in the case of a ®xed boundary) to that appropriate for an ambient (inviscid) ¯ow. In theory, as well as in practice, the approach to ambient velocity is often very smooth and asymptotic, so that some arbitrariness or ambiguity is always involved in de®ning the boundary layer thickness. In engineering practice, the outer edge of the boundary layer is usually taken where the mean velocity has attained 99% of its ambient value. This de®nition is found to be quite unsatisfactory in meteorological applications, where other more useful de®nitions of the boundary layer thickness have been used (e.g., the Ekman layer thickness de®ned earlier). Prandtl's boundary layer hypothesis is found to be valid for laminar as well as turbulent boundary layers. The fact that the boundary layer is thin compared with the distance over which it develops along a boundary allows for certain approximations to be made in the equations of motion. These boundary layer approximations, also due to Prandtl, amount to the following simpli®cations for the viscous di€usion terms in Equation (7.4): nH2 u % n…@ 2 u=@z2 † nH2 v % n…@ 2 v=@z2 † 2

2

…7:32†

2

nH w % n…@ w=@z † which follow from the neglect of velocity gradients parallel to the boundary in comparison to those normal to it.

7.6.1 The ¯at-plate laminar boundary layer Let us consider the simple case of a steady, two-dimensional boundary layer developing over a thin ¯at plate placed in an otherwise steady, uniform stream of ¯uid, with streamlines of the ambient ¯ow parallel to the plate (see Figure 7.4). This is an important classical case of ¯uid ¯ow which provides a standard

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7 An Introduction to Viscous Flows

Figure 7.4 Schematic of a developing boundary layer over a ¯at plate.

for comparison with boundary layers developing on slender bodies, such as aircraft wings, ship hulls, and turbine blades. Except in a small region near the edge of the plate, where boundary layer approximations may not be valid, the relevant boundary layer equations to be solved are u…@u=@x† ‡ w…@u=@z† ˆ n…@ 2 u=@z2 † @u=@x ‡ @w=@z ˆ 0

…7:33†

with the boundary conditions u = w = 0, at z = 0 u ? U?, as z ? ? u = U?, at x = 0, for all z where U? is the uniform ambient velocity just outside the boundary layer. Some idea about the growth of the boundary layer with distance downstream of the leading edge can be obtained by requiring that the inertial and viscous terms of Equation (7.32) be of the same order of magnitude in the boundary layer, i.e., u(@u/@x) * n(@ 2u/@z2) or U2?/x * nU?/h2 or h * (nx/U?)1/2 = x/Re1/2 x

(7.34)

7.6 Developing Laminar Boundary Layers

135

where Rex = U?x/n is a local Reynolds number. The above result indicates that the boundary thickness grows in proportion to the square root of the distance and that the ratio h/x decreases inversely proportional to the square root of the local Reynolds number. This supports the idea of a thin boundary layer in a large Reynolds number ¯ow. Equation (7.32) can be solved numerically. The resulting velocity pro®les at various distances from the edge of the plate turn out to be similar in the sense that they collapse onto a single curve if the normalized longitudinal velocity u/U? is plotted as function of the normalized distance z/h, or z/(nx/U?)1/2, from the surface. If, to begin with, one assumes a similarity solution of the form u/U? = f '(Z) = @f/@Z

(7.35)

where Z = z(U?/nx)1/2, it is easy to show that the partial di€erential equations [Equation (7.33)] yield a single ordinary di€erential equation f 000 ‡ 12 ff 00 ˆ 0

…7:36†

where prime denotes di€erentiation with respect to Z. The boundary conditions can be transformed to f = f ' = 0, at Z = 0 f ' ? 1, as Z ? ? The normalized velocity pro®le obtained from the numerical solution of the above equations is shown in Figure 7.5. Note that the pro®le near the plate surface is nearly linear, similar to the velocity pro®les in Ekman layers and plane-parallel ¯ows (this linearity of pro®les seems to be a common feature of all laminar ¯ows). The shear stress on the plate surface is given by t0 ˆ m…@u=@z†zˆ0 ˆ 0:33rU21 Rex 1=2

…7:37†

according to which the friction or drag coecient, CD : t0/rU2?, varies inversely proportional to the square root of the local Reynolds number. The boundary layer thickness (de®ned as the value of z where u = 0.99U?) is given by h % 5(nx/U?)1/2

(7.38)

Many measurements in ¯at-plate laminar boundary layers have con®rmed these theoretical results (Schlichting, 1960).

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7 An Introduction to Viscous Flows

Figure 7.5 Comparison of theoretical and observed velocity pro®les in the laminar ¯atplate boundary layer. [From Schlichting (1960).]

7.7 Heat Transfer in Laminar Flows In Section 4.4 we derived the one-dimensional equation of heat conduction in a solid medium and pointed out how it can be generalized to three dimensions. Heat transfer in a still ¯uid is no di€erent from that in a solid medium, molecular conduction or di€usion being the only mechanism of transfer. In a moving ¯uid, however, heat is more eciently transferred by ¯uid motions while thermal conduction is a relatively slow process. 7.7.1 Forced convection Consideration of conservation of energy in an elementary ¯uid volume, in a Cartesian coordinate system, leads to the following equation of heat transfer in a low-speed, incompressible ¯ow in which temperature or heat is considered only a passive admixture not a€ecting in any way the dynamics. DT/Dt = ahH2T

(7.39)

where ah is the molecular thermal di€usivity of the ¯uid and the total derivative has the usual meaning as de®ned by Equation (7.5) and incorporates the important contribution of ¯uid motions.

7.7 Heat Transfer in Laminar Flows

137

Equation (7.39) describes the variation of temperature for a given velocity ®eld and can be solved for the appropriate boundary conditions and ¯ow situations. Analytical solutions are possible only for certain types of laminar ¯ows. For two-dimensional thermal boundary layers, jets, and plumes the boundary layer approximations are used to simplify Equation (7.39) to the form u(@T/@x) + w(@T/@z) = ah(@ 2T/@z2)

(7.40)

which is similar to the simpli®ed boundary layer equation of motion. Analogous to the Reynolds number, one can de®ne the Peclet number Pe = UL/ah, which represents the ratio of the advection terms to the molecular di€usion terms in the energy equation. Instead of Pe, it is often more convenient to use the Prandtl number Pr = n/ah, which is a property of the ¯uid only and not of the ¯ow; note that Pe = Re 6 Pr

(7.41)

For air and other diatomic gases, Pr % 0.7 and is nearly independent of temperature, so that the Peclet number is of the same order of magnitude as the Reynolds number. The di€erence in the temperature at a point in the ¯ow and at the boundary is usually normalized by the temperature di€erence DT across the entire thermal layer and is represented as a function of normalized coordinates of the point, the appropriate Reynolds number, and the Prandtl number. The heat ¯ux on a boundary is usually represented by the dimensionless ratio, called the Nusselt number Nu = H0L/kDT

(7.42)

CH = H0/rcpUDT

(7.43)

or, the heat transfer coecient

which is also called the Stanton number. Note that the above parameters are related (through their de®nitions) as CH = Nu/RePr = Nu/Pe

(7.44)

Many calculations, as well as observations of heat transfer in pipes, channels, and boundary layers have been used to establish empirical correlations between the Nusselt number or the heat transfer coecient as a function of the Reynolds and Prandtl numbers. Some of these have found practical applications in

138

7 An Introduction to Viscous Flows

micrometeorological problems, particularly those dealing with heat and mass transfer to or from individual leaves (Lowry, 1970; Monteith, 1973; Gates, 1980; Monteith and Unsworth, 1990).

7.7.2 Free convection In free or natural convection ¯ows, temperature cannot be considered as a passive admixture, because inhomogeneities of the temperature ®eld in the presence of gravity give rise to signi®cant buoyant accelerations that a€ect the dynamics of the ¯ow, particularly in the vertical direction. This is usually the case in the atmosphere. The equations of motion and thermodynamic energy are slightly modi®ed to allow for the buoyancy e€ects of temperature or density strati®cation. It is generally assumed that there is a reference state of the atmosphere at rest characterized by temperature T0, density r0, and pressure p0, which satisfy the hydrostatic equation @p0/@z =7r0g

(7.45)

and that, in the actual atmosphere, the deviations in these properties from their reference values (T1 = T 7 T0, r1 = r 7 r0 and p1 = p 7 p0) are small compared to the values for the reference atmosphere (T1  T0, r1  r0, etc). These Boussinesq assumptions lead to the approximation (1/r)(@p/@z) + g % (1/r0)(@p1@z) + (g/r0)r1

(7.46)

which can be used in the vertical equation of motion. In an ideal gas, such as air, changes in density are simply related to changes in temperature as r1 =7br0T1 %7(r0/T0)T1

(7.47)

in which the coecient of thermal expansion b =7(1/r0)(@r/@T)r % 1/T0. After substituting from Equations (7.46) and (7.47) in the equation of vertical motion [Equation (7.4)], we have Dw/Dt =7 (1/r0)(@p1/@z) + (g/T0)T1 + nH2w

(7.48)

Here, the second term on the right-hand side is the buoyant acceleration due to the deviation of temperature from the reference state. The equations of horizontal motion remain unchanged; alternatively, one can replace the pressure gradient terms (1/r)(@p/@x) and (1/r)(@p/@y) by (1/r0)(@p1/@x) and

Problems and Exercises

139

(1/r0)(@p1/@y), respectively. A more appropriate form of the thermodynamic energy equation, for micrometeorological applications, is Dy/Dt = ahH2y

(7.49)

in which y is the potential temperature. The above equations are valid for both stable and unstable strati®cations. In the particular case of true free convection over a ¯at plate (horizontal or vertical) in which the motions are entirely generated by surface heating, the relevant dimensionless parameters on which temperature and velocity ®elds depend are the Prandtl number and the Grashof number Gr = (g/T0)(L3DT/n2)

(7.50)

Instead of Gr, one may also use the Rayleigh number Ra = (g/T0)(L3DT/nah) = Gr 6 Pr

(7.51)

Both the Grashof and Rayleigh numbers are expected to be large in the atmosphere, although true free convection in the sense of no geostrophic wind forcing may be a rare occurrence. 7.8 Applications Since micrometeorology deals primarily with the phenomena and processes occurring within the atmospheric boundary layer, some familiarity with the fundamentals of viscous and inviscid ¯ows is essential. In particular, the basic di€erences between viscous and inviscid ¯ows and between laminar and turbulent viscous ¯ows should be recognized. The Navier±Stokes equations of motion and the thermodynamic energy equation and diculties of their solution must be familiar to the students of micrometeorology. Introduction to some of the fundamentals of ¯uid ¯ow and heat transfer may provide a useful link between dynamic meteorology and ¯uid mechanics. A direct application of this might be in micrometeorological studies of momentum, heat, and mass transfer between the atmosphere and the physical and biological elements (e.g., snow, ice, and water surfaces and plant leaves, animals, and organisms). Problems and Exercises 1. In what regions or situations in the atmosphere may the inviscid ¯ow theory not be applicable and why?

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7 An Introduction to Viscous Flows

2. What are the basic diculties in the solution of the Navier±Stokes equations of motion for laminar viscous ¯ows? 3. Distinguish between the following types of environmental ¯uid ¯ows, giving an example of each type from the atmosphere: (a) inviscid and viscous ¯ows; (b) laminar and turbulent ¯ows; (c) forced and free convection. 4. In the gravity ¯ow down an inclined plane, determine the volume ¯ow rate Q across the plane per unit width of the plane and express the layer depth h as a function of Q and the slope angle b. 5. (a) What are the various limitations of the classical Ekman-layer solution when compared to observations in the atmospheric boundary layer. (b) Discuss di€erent ways of improving the theoretical solution to the equation of motion for the PBL. 6. Using the solution to the equations of mean horizontal motion in the PBL given in the worked-out Example Problem 2, calculate and plot the following: (a) The cross-isobar angle near the surface as a function of the ratio Us/G. (b) Pro®les of normalized velocity components U/G and V/G as functions of height z up to 1000 m for the parametric values given in the example problem, which are typical of a neutral PBL. (c) Velocity component pro®les for a stable boundary layer of height h = 100 m and Us/G = 0.35, using the appropriate value of K for the given h. Compare these with those for the neutral case in (b). 7. (a) By direct substitution verify that Equation (7.29) is a solution to Equation (7.27) which satis®es the boundary conditions [Equations (7.28)]. (b) Using the above solution [Equation (7.29)], obtain the corresponding expressions for the horizontal shear stress components tzx and tzy. (c) Show that in a coordinate system with the x axis parallel to the surface shear p stress (this t0 and tzy = 0, at z = 0), Ug = G/ 2, pimplies that tzx = Vg =7G/ 2, and t0 = rG(nf )1/2. (d) In the same coordinate system write down the expressions for the normalized velocity components (u/G and v/G) and the normalized shear stress components (tzx/t0 and tzy/t0). (e) Using the above expressions, calculate and plot the vertical pro®les of the normalized velocity and shear stress components as functions of az from 0 to 2p.

Problems and Exercises

141

(f ) What conclusions can you draw from the above pro®les? Comment on their applicability to the real atmosphere, if n can be replaced by an e€ective viscosity K. 8. Starting from the equations of motion with boundary-layer approximations, derive Equation (7.36) for the dimensionless velocity pro®le in a ¯at-plate boundary layer. 9. (a) What Boussinesq assumptions or approximations are commonly used to simplify the equations of motion in a thermally strati®ed environment. (b) Derive the approximate expression (7.46) and discuss its usefulness and application to modeling small-scale atmospheric ¯ows.

Chapter 8

Fundamentals of turbulence

8.1 Instability of Flow and Transition to Turbulence 8.1.1 Types of instabilities In Chapter 5 we introduced the concept of static (gravitational) stability or instability and showed how the parameter s = (g/Tv)(@Yv/@z %7(g/r)(@r/@z) may be used as a measure of static stability of an atmospheric (¯uid) layer. This was based on the simple criterion whether vertical motions of ¯uid parcels were suppressed or enhanced by the buoyancy force arising from density di€erences between the parcel and the environment. In particular, when s 5 0, the ¯uid layer is gravitationally unstable; the parcel moves farther and farther away from its equilibrium position. Another type of ¯ow instability is the dynamic or hydrodynamic instability. A ¯ow is considered dynamically stable if perturbations introduced in the ¯ow, either intentionally or inadvertently, are found to decay with time or distance in the direction of ¯ow, and eventually are suppressed altogether. It is dynamically unstable if the perturbations grow in time or space and irreversibly alter the nature of the basic ¯ow. A ¯ow may be stable with respect to very small (in®nitesimal) perturbations but might become unstable if large (®nite) amplitude disturbances are introduced in the ¯ow. A simple example of dynamically unstable ¯ow is that of two inviscid ¯uid layers moving initially at di€erent velocities so that there is a sharp discontinuity at the interface. The development of this inviscid shear instability can be demonstrated by assuming a slight perturbation (e.g., in the form of a standing wave) of the interface and the resulting perturbations in pressure (in accordance with the Bernoulli equation u2/2 + p/r = constant) at various points across the interface. The forces induced by pressure perturbations tend to progressively increase the amplitude of the wave, no matter how small it was to begin with, thus irreversibly modifying the interface and the ¯ow in its vicinity (Tritton, 1988, Chapter 17). Shear instability at the interface between two parallel streams of di€erent velocities and densities, with the heavier ¯uid at the bottom, or of continuously strati®ed ¯ows is called the Kelvin±Helmholtz (KH) instability. 142

8.1 Instability of Flow and Transition to Turbulence

143

The KH instability is caused by the destabilizing e€ect of shear, which overcomes the stabilizing e€ect of strati®cation. This type of instability is ubiquitous in the atmosphere and oceans and is considered to be a major source of internal gravity waves in these environments (Kundu, 1990, Chapter 11). In real ¯uids, of course, the processes leading to the development of mixing layers are more complicated, but the initiation of the Kelvin±Helmholtz instability at the interface is qualitatively similar to that in an inviscid ¯uid. Laminar ¯ows in channels, tubes, and boundary layers, introduced in Chapter 7, are found to be dynamically stable (i.e., they can be maintained as laminar ¯ows) only under certain restrictive conditions (e.g., Reynolds number less than some critical value Rec and a relatively disturbance-free environment), which can be realized only in carefully controlled laboratory facilities. For example, by carefully eliminating all extraneous disturbances, laminar ¯ow in a smooth tube has been realized up to Re % 105, while under ordinary circumstances the critical Reynolds number (based on mean velocity and tube diameter) is only about 2000. Similarly, the critical Reynolds numbers for the instability of laminar channel and boundary layer ¯ows are also of the same order of magnitude. For other ¯ows, such as jets and wakes, instabilities occur at much lower Reynolds numbers. Thus, all laminar ¯ows are expected to become dynamically unstable and, hence, cannot be maintained as such, at suciently large Reynolds numbers (Re 4 Rec) and in the presence of disturbances ordinarily present in the environment (Kundu, 1990, Chapter 11). The above Reynolds number criterion for dynamic stability does not mean that all inviscid ¯ows are inherently unstable, because zero viscosity implies an in®nite Reynolds number. Actually, many nonstrati®ed (uniform density) inviscid ¯ows are found to be stable with respect to small perturbations, unless their velocity pro®les have discontinuities or in¯exion points. The development of Kelvin±Helmholtz instability along the plane of discontinuity in the velocity pro®le has already been discussed. Even without a sharp discontinuity, the existence of an in¯exion point in the velocity pro®le constitutes a necessary condition for the occurrence of instability in a neutrally strati®ed ¯ow; it is also a sucient condition for the ampli®cation of disturbances. This inviscid instability mechanism also operates in viscous ¯ows having points of in¯exion in their velocity pro®les. It is for this reason that laminar jets, wakes, mixing layers, and separated boundary layers become unstable at low Reynolds numbers. The laminar Ekman layers are also inherently unstable because their velocity pro®les have in¯exion points. In all such ¯ows, both the viscous and inviscid shear instability mechanisms operate simultaneously and consequently, the stability criteria are more complex (Kundu, 1990, Chapter 11). Gravitational instability is another mechanism through which both the inviscid and viscous streamlined ¯ows can become turbulent. Flows with ¯uid density increasing with height become dynamically unstable, particularly at

144

8 Fundamentals of turbulence

large Reynolds numbers (Re 4 Rec) or Rayleigh numbers (Ra 4 Rac). In the lower atmosphere, this criterion can essentially be expressed in terms of virtual potential temperature decreasing with height, because Reynolds and Rayleigh numbers are much larger than their estimated critical values. Any statically unstable layers, based on either local or nonlocal stability criteria, are also expected to be dynamically unstable and, hence, generally turbulent. The daytime unstable or convective boundary layer is a prime example of gravitational instability. Stably strati®ed ¯ows with weak negative density gradients and/or strong velocity gradients can also become dynamically unstable, if the Richardson number is less than its critical value Ric = 0.25 (this value is based on a number of theoretical as well as experimental investigations). The Richardson number criterion is found to be extremely useful for identifying possible regions of active or incipient turbulence in stably strati®ed atmospheric and oceanic ¯ows. If the local Richardson number in a certain region is less than 0.25 and there is also an in¯exion point in the velocity pro®le (e.g., in jet streams and mixing layers), the ¯ow there is likely to be turbulent. This criterion is found to be generally valid in strati®ed boundary layers, as well as in free shear ¯ows, and is often used in forecasts of clear-air turbulence in the upper troposphere and lower stratosphere for aircraft operations. Dynamic instability promotes the growth of wave-like disturbances which may break down to produce turbulence. The instability-generated wave motion may also become dynamically stable and not always result in a turbulent ¯ow. The nighttime stable boundary layer is a prime example of the shear instability of a strati®ed ¯ow. 8.1.2 Methods of investigation The stability of a physically realizable laminar ¯ow can be investigated experimentally and the approximate stability criteria can be determined empirically. This approach has been used by experimental ¯uid dynamists ever since the classical experiments in smooth glass tubes carried out by Osborne Reynolds. Systematic and carefully controlled experiments are required for this purpose (Monin and Yaglom, 1971, Chapter 1). Empirical observations to test the validity of the Richardson number criterion have also been made by meteorologists and oceanographers. Another, more frequently used approach is theoretical, in which smallamplitude wavelike perturbations are superimposed on the original laminar or inviscid ¯ow and the conditions of stability or instability are examined through analytical or numerical solutions of perturbation equations (Monin and Yaglom, 1971, Chapter 1; Kundu, 1990, Chapter 11). The analysis is straightforward and simple, so long as the amplitudes of perturbations are small enough

8.1 Instability of Flow and Transition to Turbulence

145

for the nonlinear terms in the equations to be dropped out or linearized. It becomes extremely complicated and even intractable in the case of ®niteamplitude perturbations. Thus, most theoretical investigations give only some weak criteria of the stability of ¯ow with respect to small-amplitude (in theory, in®nitesimal) perturbations. Investigations of ¯ow stability with respect to ®nite-amplitude perturbations are being attempted now using new approaches to the numerical solution of nonlinear perturbation equations on large computers. Comprehensive reviews of the literature on stability analyses of viscous and inviscid ¯ows are given by Chandrasekhar (1961), Monin and Yaglom (1971), and Drazin and Reid (1981). The importance of stability analysis arises from the fact that any solution to the equations of motion, whether exact or approximate, cannot be considered physically realizable unless it can be shown that the ¯ow represented by the solution is also stable with respect to small perturbations. If the basic ¯ow is found to be unstable in a certain range of parameters, it cannot be realized under those conditions, because naturally occurring in®nitesimal disturbances will cause instabilities to develop and alter the ¯ow ®eld irreversibly. 8.1.3 Transition to turbulence Experiments in controlled laboratory environments have shown that several distinctive stages are involved in the transition from laminar ¯ow (e.g., the ¯atplate boundary layer) to turbulence. The initial stage is the development of primary instability which, in simple cases, may be two-dimensional. The primary instability produces secondary motions which are generally threedimensional and become unstable themselves. The subsequent stages are the ampli®cation of three-dimensional waves, the development of intense shear layers, and the generation of high-frequency ¯uctuations. Finally, `turbulent spots' appear more or less randomly in space and time, grow rapidly, and merge with each other to form a ®eld of well-developed turbulence. The above stages are easy to identify in a ¯at-plate boundary layer and other developing ¯ows, because changes occur as a function of increasing Reynolds number with distance from the leading edge. In fully developed two-dimensional channel and pipe ¯ows, however, transition to turbulence occurs more suddenly and explosively over the whole length of the channel or pipe as the Reynolds number is increased beyond its critical value. Mathematically, the details of transition from initially laminar or inviscid ¯ow to turbulence are rather poorly understood. Much of the theory is linearized, valid for small disturbances, and cannot be used beyond the initial stages. Even the most advanced nonlinear theories dealing with ®nite-amplitude disturbances cannot handle the later stages of transition, such as the develop-

146

8 Fundamentals of turbulence

ment of `turbulent spots.' A rigorous mathematical treatment of transition from turbulent to laminar ¯ow is also lacking, of course, due to the lack of a generally valid and rigorous theory or model of turbulence. Some of the turbulent ¯ows encountered in nature and technology may not go through a transition of the type described above and are produced as such (turbulent). Flows in ordinary pipes, channels, and rivers, as well as in atmospheric and oceanic boundary layers, are some of the commonly occurring examples of such ¯ows. Other cases, such as clear air turbulence in the upper atmosphere and patches of turbulence in the strati®ed ocean, are obviously the results of instability and transition processes. Transitions from laminar ¯ow to turbulence and vice versa continuously occur in the upper parts of the stable boundary layer, throughout the night. Micrometeorologists have a deep and abiding interest in understanding these transition processes, because turbulent exchanges of momentum, heat and mass in the SBL and other stable layers occur intermittently during episodes of turbulence generation. 8.2 The Generation and Maintenance of Turbulence It is not easy to identify the origin of turbulence. In an initially nonturbulent ¯ow, the onset of turbulence may occur suddenly through a breakdown of streamlined ¯ow in certain localized regions (turbulent spots). The cause of the breakdown, as pointed out in the previous section, is an instability mechanism acting upon the naturally occurring disturbances in the ¯ow. Since turbulence is transported downstream in the manner of any other ¯uid property, repeated breakdowns are required to maintain a continuous supply of turbulence, and instability is an essential part of this process. Once turbulence is generated and becomes fully developed in the sense that its statistical properties achieve a steady state, the instability mechanism is no longer required (although it may still be operating) to sustain the ¯ow. This is particularly true in the case of shear ¯ows, where shear provides an ecient mechanism for converting mean ¯ow energy to turbulence kinetic energy (TKE). Similarly, in unstably strati®ed ¯ows, buoyancy provides a mechanism for converting potential energy of strati®cation into turbulence kinetic energy (the reverse occurs in stably strati®ed ¯ows). The two mechanisms for turbulence generation become more evident from the so-called shear production (S) and buoyancy production (B) terms in the TKE equation d(TKE)/dt = S + B 7 D + Tr

(8.1)

which is written here in a short-form notation for the various terms; a more complete version is given later in Chapter 9. The TKE equation also shows

8.3 General Characteristics of Turbulence

147

that there is a continuous dissipation (D) of energy by viscosity in any turbulent ¯ow and there may be transport (Tr) of energy from or to other regions of ¯ow. Thus, in order to maintain turbulence, one or more of the generating mechanisms must be active continuously. For example, in the daytime unstable or convective boundary layer, turbulence is produced both by shear and buoyancy (the relative contribution of each depends on Richardson number). Shear is the only e€ective mechanism for producing turbulence in the nocturnal stable boundary layer, in low-level jets, and in regions of the upper troposphere and stratosphere where negative buoyancy actually suppresses turbulence. Near the earth's surface wind shears become particularly intense and e€ective, because wind speed must vanish at the surface and air ¯ow has to go around the various surface inhomogeneities. For this reason, shear-generated turbulence is always present in the atmospheric surface layer. Richardson (1920) originally proposed a particularly simple condition for the maintenance of turbulence in stably strati®ed ¯ows, viz., that the rate of production of turbulence by shear must be equal to or greater than the rate of destruction by buoyancy. That is, the Richardson number, which turns out to be the ratio of the buoyancy destruction to shear production, must be smaller than or equal to unity. An important assumption implied in deriving this condition was that there is no viscous dissipation of energy on reaching the critical condition for the sudden decay of turbulence. On the other hand, subsequent observations of the transition from turbulent to laminar ¯ow have indicated viscous dissipation to remain signi®cant; they also suggest a much lower value of the critical Richardson number (Ric), in the range of 0.2 to 0.5 [it has not been possible to determine Ric very precisely (see Arya, 1972)]. There is no requirement that this should be the same as the critical Richardson number (Ric = 1/4) for the transition from laminar to turbulent ¯ow, which has been determined more rigorously and precisely from theoretical stability analyses. There is some observational evidence that the critical Richardson number for the transition from turbulent to laminar ¯ow increases with height in the SBL and may be close to one at the top of the SBL.

8.3 General Characteristics of Turbulence Even though turbulence is a rather familiar notion, it is not easy to de®ne precisely. In simplistic terms, we refer to very irregular and chaotic motions as turbulent. But some wave motions (e.g., on an open sea surface) can be very irregular and nearly chaotic, but they are not turbulent. Perhaps it would be more appropriate to mention some general characteristics of turbulence.

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8 Fundamentals of turbulence

1. Irregularity or randomness. This makes any turbulent motion essentially unpredictable. No matter how carefully the conditions of an experiment are reproduced, each realization of the ¯ow is di€erent and cannot be predicted in detail. The same is true of the numerical simulations (based on Navier±Stokes equations), which are found to be highly sensitive to even minute changes in initial and boundary conditions. For this reason, a statistical description of turbulence is invariably used in practice. 2. Three-dimensionality and rotationality. The velocity ®eld in any turbulent ¯ow is three-dimensional (we are excluding here the so-called two-dimensional or geostrophic turbulence, which includes all large-scale atmospheric motions) and highly variable in time and space. Consequently, the vorticity ®eld is also three-dimensional and ¯ow is highly rotational. 3. Di€usivity or ability to mix properties. This is probably the most important property, so far as applications are concerned. It is responsible for the ecient di€usion of momentum, heat, and mass (e.g., water vapor, CO2, and various pollutants) in turbulent ¯ows. Macroscale di€usivity of turbulence is usually many orders of magnitude larger than the molecular di€usivity. The former is a property of the ¯ow while the latter is a property of the ¯uid. Turbulent di€usivity is largely responsible for the evaporation in the atmosphere, as well as for the spread (dispersion) of pollutants released in the atmospheric boundary layer. It is also responsible for the increased frictional resistance of ¯uids in pipes and channels, around aircraft and ships, and on the earth's surface. 4. Dissipativeness. The kinetic energy of turbulent motion is continuously dissipated (converted into internal energy or heat) by viscosity. Therefore, in order to maintain turbulent motion, the energy has to be supplied continuously. If no energy is supplied, turbulence decays rapidly. 5. Multiplicity of scales of motion. All turbulent ¯ows are characterized by a wide range (depending on the Reynolds number) of scales or eddies. The transfer of energy from the mean ¯ow into turbulence occurs at the upper end of scales (large eddies), while the viscous dissipation of turbulent energy occurs at the lower end (small eddies). Consequently, there is a continuous transfer of energy from the largest to the smallest scales. Actually, it trickles down through the whole spectrum of scales or eddies in the form of a cascade process. The energy transfer processes in turbulent ¯ows are highly nonlinear and are not well understood. Of the above characteristics, rotationality, di€usivity, and dissipativeness are the properties which distinguish a three-dimensional random wave motion from turbulence. The wave motion is nearly irrotational, nondi€usive, and nondissipative.

8.4 Mean and Fluctuating Variables

149

8.4 Mean and Fluctuating Variables In a turbulent ¯ow, such as the PBL, velocity, temperature and other variables vary irregularly in time and space; it is therefore common practice to consider these variables as sums of mean and ¯uctuating parts, e.g., u~ ˆ U ‡ u v~ ˆ V ‡ v ~ ˆW‡w w ~ˆY‡y y

…8:2†

in which the left-hand side represents an instantaneous variable (denoted by a tilde) and the right-hand side its mean (denoted by a capital letter) and ¯uctuating (denoted by a lower case letter) parts. The decomposition of an instantaneous variable in terms of its mean and ¯uctuating parts is called Reynolds decomposition, because it was ®rst proposed by Reynolds (1894). There are several types of means or averages used in theory and practice. The most commonly used in the analysis of observations from ®xed instruments is the time mean, which, for a continuous record (time series), can be de®ned as Fˆ

1 T

Z 0

T

~ f…t†dt

where f~ is any variable or a function of variables, F is the corresponding time mean, and T is the length of record or sampling time over which averaging is desired. For the digitized data, mean is simply an arithmetic average of all the digitized values of the variable during the chosen sampling period. In certain ¯ows, such as the atmospheric boundary layer, the choice of an optimum sampling time or the averaging period T is not always clear. It should be suciently long to ensure stable averages and to incorporate the e€ects of all the signi®cantly contributing large eddies in the ¯ow. On the other hand, T should not be too long to mask what may be considered as real trends (e.g., the diurnal variations) in the ¯ow. In the analysis of micrometeorological observations, the optimum averaging time may range between 103 and 104 s, depending on the height of observation, the PBL height, and stability. Time averages are frequently used for micrometeorological variables measured by instruments placed on the ground or mounted on masts, towers, or tethered balloons. Another type of averaging used, particularly in the analyses of aircraft, radar, and sodar observations, is the spatial average of all the observations over the covered space. The optimum length, area, or volume over which the spatial averaging is performed depends on the spatial scales of signi®cant large eddies

150

8 Fundamentals of turbulence

in the ¯ow. Horizontal line and area averages are more meaningful, considering the inhomogeneity of the PBL in the vertical due to shear and buoyancy e€ects. Finally, the type of averaging which is almost always used in theory, but rarely in practice, is the ensemble or probability mean. It is an arithmatical average of a very large (approaching in®nity) number of realizations of a variable or a function of variables, which are obtained by repeating the experiment over and over again under the same general conditions. It is quite obvious that the ensemble averages would be nearly impossible to obtain under the varying weather conditions in the atmosphere, over which we have little or no control. Even in a controlled laboratory environment, it would be very time consuming to repeat an experiment many times to obtain ensemble averages. The fact that di€erent types of averages are used in theory and experiments requires that we should know about the conditions in which time or space averages might become equivalent to ensemble averages. It has been found that the necessary and sucient conditions for the time and ensemble means to be equal are that the process (in our case, ¯ow) be stationary (i.e., the averages be independent of time) and the averaging period be very large (T ? ?) (for a more detailed discussion of these conditions, see Monin and Yaglom, 1971). It should suce to say that these conditions cannot be strictly satis®ed in the atmosphere so that the above-mentioned equivalence of averages can only be approximate. The corresponding conditions for the equivalence of spatial and ensemble averages are spatial homogeneity (independence of averages to spatial coordinates in one or more directions) and very large averaging paths or areas. These conditions are even more restrictive and dicult to satisfy in micrometeorological applications. Quasi-stationarity over a limited period of time, or quasi-homogeneity over a limited length or area in the horizontal plane, is the best one can hope for in an idealized PBL over a homogeneous surface under undisturbed weather conditions. Under such conditions, one might expect an approximate correspondence between time or space averages used in experiments and observations and ensemble averages used in theories and mathematical models of the PBL. Irrespective of the type of averaging used, it is obvious from Reynolds decomposition that a ¯uctuation is the deviation of an instantaneous variable from its mean. By de®nition, then, the mean of any ¯uctuating variable is zero ( u ˆ 0, v ˆ 0, etc.), so that there are compensating negative ¯uctuations for positive ¯uctuations on the average. Here, an overbar over a variable denotes its mean. The relative magnitudes of mean and ¯uctuating parts of variables in the atmospheric boundary layer depend on the type of variable, observation height relative to the PBL height, atmospheric stability, the type of surface, and other factors. In the surface layer, during undisturbed weather conditions, the magnitudes of vertical ¯uctuations, on the average, are much larger than the

8.4 Mean and Fluctuating Variables

151

mean vertical velocity, the magnitudes of horizontal velocity ¯uctuations are of the same order or less than the mean horizontal velocity, and the magnitudes of the ¯uctuations in thermodynamic variables are at least two orders of magnitude smaller than their mean values. The relative magnitudes of turbulent ¯uctuations generally decrease with increasing stability and also with increasing height in the PBL. As an example, measured time series of velocity, temperature, and humidity ¯uctuations at a suburban site in Vancouver, Canada, during the daytime moderately unstable conditions are shown in Figure 8.1. These observations were taken at a height of 27.4 m above ground level, which falls within the unstable surface layer. The corresponding hourly averaged values of the variables were: U = 3.66 m s71; V = W = 0 T = 294.6 K; Q = 8.3 g m73 Note the di€erence in the character of traces of horizontal and vertical velocity components, temperature, and absolute humidity ¯uctuations. Under unstable and convective conditions represented by these traces, buoyant plumes and thermals cause strong asymmetry between positive and negative

Figure 8.1 Observed time series of velocity, temperature, and absolute humidity ¯uctuations in the atmospheric surface layer at a suburban site in Vancouver, Canada, during moderately unstable conditions. [From Roth (1991).]

152

8 Fundamentals of turbulence

¯uctuations, particularly in vertical velocity, temperature, and humidity. Under strong winds and near-neutral stability, on the other hand, ¯uctuations of the various variables tend to be more or less similar, with an approximate symmetry between positive and negative ¯uctuations. 8.5 Variances and Turbulent Fluxes The vertical pro®les of mean variables, such as mean velocity components, temperature, etc., can tell much about the mean structure of ¯ow, but little or nothing about the turbulent structure and the various exchange processes taking place in the ¯ow. Various statistical measures are used to study and represent the turbulence structure. These are all based on statistical analyses of turbulent ¯uctuations observed in the ¯ow. The simplest measures of ¯uctuation levels are the variances u2 , v2 , w2 [these three are often combined to de®ne the turbulence kinetic energy per unit mass (1/2)(u2 + v2 + w2 )],  y2, etc., and standard deviations su ˆ …u2 †1=2 , etc. The ratios of standard deviations of velocity ¯uctuations to the mean wind speed are called turbulence intensities (e.g., iu = su/|V|, iv = sv/|V| and iw = sw/|V|), which are measures of relative ¯uctuation levels in di€erent directions (velocity components). Note that the mean wind speed rather than the particular component velocity, is used in the de®nition of turbulence intensities. When the x axis is oriented with the mean wind, iu, iv, and iw are designated as longitudinal, lateral, and vertical turbulence intensities, respectively. Observations indicate that turbulence intensities are typically less than 10% in the nocturnal boundary layer, 10±15% in a near-neutral surface layer, and greater than 15% in unstable and convective boundary layers. Turbulence intensities are generally largest near the surface and decrease with height in stable and near-neutral boundary layers. Under convective conditions, however, the secondary maxima in turbulence intensities often occur at the inversion base and sometimes in the middle of the CBL. Even more important in turbulent ¯ows are the covariances uw, vu, yw, etc., which are directly related to and sometimes referred to as turbulent ¯uxes of momentum, heat, etc. As covariances, they are averages of products of two ¯uctuating variables and depend on the correlations between the variables involved. These can be positive, negative, or zero, depending on the type of ¯ow and symmetry conditions. For example, if u and w are the velocity ¯uctuations in the x and z directions, respectively, their product uw will also be a ¯uctuating quantity but with a nonzero mean; uw will be positive if u and w are positively correlated, and negative if the two variables are negatively correlated. A casual inspection of u and w time series in Figure 8.1 indicates that these variables are negatively correlated, so that uw 5 0.

8.5 Variances and Turbulent Fluxes

153

A better measure of the correlation between two variables is provided by the correlation coecient ruw ˆ uw=su sw

…8:3†

whose value always lies between 71 and 1, according to Schwartz's inequality |uw| 4 susw. Similarly, one can de®ne correlation coecients for all the covariances to get an idea of how well correlated di€erent variables are in a turbulent ¯ow. For example, observations in Figure 8.1 indicate that temperature ¯uctuations are well correlated with vertical velocity ¯uctuations with rwy % 0.6. There is still some order to be found in an apparently chaotic motion which is responsible for all the important exchange processes taking place in the ¯ow. In order to see a clear connection between covariances and turbulent ¯uxes, let us consider the vertical ¯ux of a scalar with a variable concentration c~ in the ¯ow. The ¯ux of any scalar in a given direction is de®ned as the amount of the scalar per unit time per unit area normal to that direction. It is quite obvious that the velocity component in the direction of ¯ux is responsible for the transport and hence for the ¯ux. For example, it is easy to see that in the vertical ~ and the mean ¯ux with which we are direction the ¯ux at any instant is c~w ~ Further writing c~ ˆ C ‡ c and w ~ ˆ W ‡ w and normally concerned is c~w. following the Reynolds averaging rules, it can be shown that ~ ˆ …C ‡ c†…W ‡ w† ˆ CW ‡ cw c~w

…8:4†

Thus, the total scalar ¯ux can be represented as a sum of the mean transport (transport by mean motion) and the turbulent transport. The latter, also called the turbulent ¯ux, is usually the dominant transport term and always of considerable importance in a turbulent ¯ow. If the scalar under consideration is the potential temperature ~ y, or enthalpy rcp ~y, the corresponding turbulent ¯ux in the vertical is yw, or rcp yw. In this way, the covariance yw may be interpreted as the turbulent heat ¯ux (in kinematic units). Similarly, qw may be considered as the vertical turbulent ¯ux of moisture or water vapor. Turbulent ¯uxes of momentum involve covariances between velocity ¯uctuations in di€erent directions. For example, by symmetry, ruw may be considered as the vertical ¯ux of u momentum and also as the horizontal (x direction) ¯ux of the vertical momentum. Since the rate of change (¯ux) of momentum is equal to the force per unit area, or the stress, turbulent ¯uxes of momentum can also be interpreted as turbulent stresses, that is, tzx ˆ

rwu;

tzy ˆ

rwv; etc:

(8.5)

154

8 Fundamentals of turbulence

Like the viscous stresses represented in Equation (7.2), the turbulent stresses above are designated by two direction subscripts. The ®rst subscript to t denotes the direction normal to the plane of the stress and the second subscript denotes the direction of the stress. There are, in all, nine stress components acting on any cubical ¯uid element, of which three (txx , tyy , and tzz ) are normal stresses, which are proportional to velocity variances, and six (txy , txz , etc.) are shearing stresses which are proportional to covariances. When expressed in terms of variances and covariances of turbulent velocity ¯uctuations, these are called Reynolds stresses. These turbulent stresses are found to be much larger in magnitude than the corresponding mean viscous stresses Tzx

    @U @W @V @W ˆm ‡ ; Tzy ˆ m ‡ ; etc: @z @x @z @y

…8:6†

These viscous stresses can usually be ignored, except within extremely thin viscous sublayers near smooth surfaces. Example Problem 1 The following mean ¯ow and turbulence measurements were made at the 22.6 m height level of the micrometeorological tower during an hour-long run of the 1968 Kansas Field Program: Mean wind speed = 4.89 m s71. Velocity variances: u2 ˆ 0:69, v2 ˆ 1:04, w2 ˆ 0:42 m2 s72. Turbulent ¯uxes: uw ˆ 0:081 m2 s72, yw = 0.185 K m s71, yu ˆ 0:064 K m s71. Calculate the following: (a) standard deviations of velocity ¯uctuations; (b) turbulence intensities; (c) correlation coecients; (d) turbulence kinetic energy (TKE) and the ratio TKE/uw. Solution (a) From the de®nition, standard deviations are su ˆ …u2 †1=2 ˆ 0:83 m s

1

sv ˆ …v2 †1=2 ˆ 1:02 m s

1

sw ˆ …w2 †1=2 ˆ 0:64 m s

1

8.6 Eddies and Scales of Motion

155

(b) The corresponding turbulence intensities can be computed as iu ˆ su =jVj ˆ 0:17 iv ˆ sv =jVj ˆ 0:21 iw ˆ sw =jVj ˆ 0:13 (c) The correlation coecients can be calculated as ruw ˆ uw=su sw ˆ

0:15

rwy ˆ yw=sy sw ˆ 0:59 ruy ˆ yu=sy su ˆ

0:16

which indicate that y and w are positively well correlated, while y and u are only weakly correlated with a negative correlation coecient of 70.16 in the convective surface layer. (d) Turbulence kinetic energy can be calculated as TKE ˆ 12…u2 ‡ v2 ‡ w2 † ˆ 1:075 m2 s TKE=uw ˆ 13:27

2

8.6 Eddies and Scales of Motion It is a common practice to speak in terms of eddies when turbulence is described qualitatively. An eddy is by no means a clearly de®ned structure or feature of the ¯ow which can be isolated and followed through, in order to study its behavior. It is rather an abstract concept used mainly for qualitative descriptions of turbulence. An eddy may be considered akin to a vortex or a whirl in common terminology. Turbulent ¯ows are highly rotational and have all kinds of vortexlike structures (eddies) buried in them. However, eddies are not simple two-dimensional circulatory motions of the type in an isolated vortex, but are believed to be complex, three-dimensional structures. Any analogy between turbulent eddies and vortices can only be very rough and qualitative. On the basis of ¯ow visualization studies, statistical analyses of turbulence data, and some of the well-accepted theoretical ideas, it is believed that a turbulent ¯ow consists of a hierarchy of eddies of a wide range of sizes (length scales), from the smallest that can survive the dissipative action of viscosity to the largest that is allowed by the ¯ow geometry. The range of eddy sizes increases with the Reynolds number of the overall mean ¯ow. In particular, for the ABL, the typical range of eddy sizes is 1073 to 103 m.

156

8 Fundamentals of turbulence

Of the wide and continuous range of scales in a turbulent ¯ow, a few have special signi®cance and are used to characterize the ¯ow itself. One is the characteristic large-eddy scale or macroscale of turbulence, which represents the length (l) or time scale of eddies receiving the most energy from the mean ¯ow. Another is the characteristic small-eddy scale or microscale of turbulence, which represents the length (Z) or time scale of most dissipating eddies. The ratio l/Z is found to be proportional to Re3/4 and is typically 105 (range: 104±106) for the atmospheric PBL. The macroscale, or simply the scale of turbulence, is generally comparable (same order of magnitude) to the characteristic scale of the mean ¯ow, such as the boundary layer thickness or channel depth, and does not depend on the molecular properties of the ¯uid. On the other hand, the microscale depends on the ¯uid viscosity n, as well as on the rate of energy dissipation e. Dimensional considerations further lead to the de®ning relationship Z : n3/4e71/4

(8.7)

In stationary or steady-state conditions, the rate at which the energy is dissipated is exactly equal to the rate at which the energy is supplied from mean ¯ow into turbulence. This leads to an inviscid estimate for the rate of energy dissipation e  u3` =l, where u` is the characteristic velocity scale of turbulence (large eddies), which can be de®ned in terms of the turbulence kinetic energy as u` ˆ

p TKE

(8.8)

Some investigators de®ne the turbulence velocity scale as the square-root of (u2 ‡ v2 ‡ w2 ) which is twice the turbulence kinetic energy (Tennekes and Lumley, 1972).

8.7 Fundamental Concepts, Hypotheses, and Theories Here, we brie¯y describe, in qualitative terms only, some of the fundamental concepts, hypotheses, and theories of turbulence, which were mostly proposed in the ®rst-half of the twentieth century and subsequently re®ned and veri®ed by experimental data on turbulence from the laboratory as well as the atmosphere. More comprehensive reviews of theories and models of turbulence are given elsewhere (Monin and Yaglom, 1971; 1975; Hinze, 1975; Panofsky and Dutton, 1984; McComb, 1990).

8.7 Fundamental Concepts, Hypotheses, and Theories

157

8.7.1 Energy cascade hypothesis One of the fundamental concepts of turbulence is that describing the transfer of energy among di€erent scales or eddy sizes. It has been recognized for a long time that in large Reynolds number ¯ows almost all of the energy is supplied by mean ¯ow to large eddies, while almost all of it is eventually dissipated by small eddies. The transfer of energy from large (energy-containing) to small (energydissipating) eddies occurs through a cascade-type process involving the whole range of intermediate size eddies. This energy cascade hypothesis was originally suggested by Lewis Richardson in 1922 in the form of a parody

Big whorls have little whorls, Which feed on their velocity; And little whorls have lesser whorls, And so on to viscosity.

This may be considered to be the qualitative picture of turbulence structure in a nutshell and the concept of energy transfer down the scale. The actual mechanism and quantitative aspects of energy transfer are very complicated and are outside the scope of this text. Smaller eddies are assumed to be created through an instability and breakdown of larger eddies and the energy transfer from larger to smaller eddies presumably occurs during the breakdown process. The largest eddies are produced by mean ¯ow shear and thermal convection. Their characteristic length scale l is of the same order as the characteristic length scale (e.g., the PBL height) of the mean ¯ow, especially for the neutral, unstable, and convective boundary layers. In the stable boundary layer, on the other hand, large-eddy size is found to be severely restricted by negative buoyancy forces, and l is usually much smaller than the PBL height h. The stronger the stability (large Ri), the smaller is the large-eddy length scale, which becomes independent of the PBL height. If the large-eddy Reynolds number Re` ˆ u` l=n is suciently large, these large eddies become dynamically unstable and produce eddies of somewhat smaller size, which themselves become unstable and produce eddies of still smaller size, and so on further down the scale. This cascade process is terminated when the Reynolds number based on the smallest eddy scales becomes small enough (order of one) for the smallest eddies to become stable under the in¯uence of viscosity. This qualitative concept of energy cascade provided an underpinning for subsequent theoretical ideas, as well as experimental studies of turbulence.

158

8 Fundamentals of turbulence

8.7.2 Statistical theory of turbulence A sophisticated mathematical and statistical theory of turbulence has been developed for the simplest type of turbulence, called homogeneous and isotropic turbulence (Hinze, 1975; Monin and Yaglom, 1975). This idealized ®eld of turbulence has no boundaries, no mean ¯ow and shear, and no thermal or density strati®cation, that may cause deviations from perfect homogeneity and isotropy. Consequently, there are no momentum, heat, and mass ¯uxes in any direction and the variance of velocity ¯uctuations is the same in all directions. In the absence of any turbulence production mechanisms (shear and buoyancy), an isotropic turbulence decays with time. Such a turbulence can be approximately realized when a uniform grid is towed through a still ¯uid, as in a laboratory water channel or a wind tunnel (uniform ¯ow passing through a grid is often used as a surrogate for a homogeneous and isotropic turbulence). Atmospheric turbulence is not homogeneous and isotropic when one considers its large-scale structure. The presence of mean wind shears and thermal strati®cation, at least in the vertical direction, are responsible for the generation and maintenance of turbulence. Horizontal inhomogeneities of mean ¯ow and turbulence are often caused by variations of surface roughness, temperature, and topography. It is not easy to extend the rigorous mathematical formulation of the statistical theory to such an inhomogeneous ¯ow and turbulence ®eld. For the horizontally homogeneous PBL, however, approximate formulations including the e€ects of vertical wind shear and buoyancy have been made, but their numerical solutions still depend on many closure hypotheses and assumptions. In short, theoretical rigor and simplicity are lost, whenever signi®cant deviations from the complete homogeneity of turbulence have to be considered.

8.7.3 Kolmogorov's local similarity theory Kolmogorov (1941) postulated that at suciently large Reynolds numbers, small-scale motions in all turbulent ¯ows have similar universal characteristics. He proposed a local similarity theory to describe these characteristics. The basic foundation of his widely accepted theory lies in the energy cascade hypothesis. Kolmogorov argued that if the characteristic Reynolds number of the mean ¯ow or that of the most energetic large eddies is suciently large, there will be many steps in the energy cascade process before the energy is dissipated by small-scale motions. The large-scale motions or eddies that receive energy directly from mean ¯ow shear or buoyancy are expected to be inhomogeneous. But the small eddies, that are formed after many successive breakdowns of large and intermediate size eddies, are likely to become homogeneous and isotropic,

8.7 Fundamental Concepts, Hypotheses, and Theories

159

because they are far enough removed from the original sources (shear and buoyancy) of inhomogeneity and have no memory of large-scale processes. Following the above reasoning, Kolmogorov (1941) proposed his local isotropy hypothesis, which states that at suciently large Reynolds numbers, small-scale structure is locally isotropic whether large-scale motions are isotropic or not. Here `local' refers to the particular range of small-scale eddy motions that may be considered isotropic. The concept of local isotropy has proved to be very useful in that it applies to all turbulent ¯ows with large Reynolds numbers. It also made possible for the suciently well-developed statistical theory of isotropic turbulence to be applied to small-scale motions in most turbulent ¯ows encountered in practice. The condition of suciently large Reynolds number is particularly well satis®ed in the atmosphere. Consequently, the local isotropy concept or hypothesis is found to be valid over a wide range of scales and eddies in the atmosphere. For locally isotropic, small-scale turbulence Kolmogorov (1941) proposed a local similarity theory or scaling based on his well-known equilibrium range hypothesis. This states that at suciently high Reynolds numbers there is a range of small scales for which turbulence structure is in a statistical equilibrium and is uniquely determined by the rate of energy dissipation (e) and kinematic viscosity (n). Here the statistical equilibrium refers to the stationarity of smallscale turbulence, even when the large-scale turbulence may not be stationary. The above two parameters lead to the following Kolmogorov's microscales from dimensional considerations: Length scale: Z = n3/4e71/4 Velocity scale: u = n1/4e1/4

(8.9)

which are used to normalize any turbulence statistics and to examine their universal similarity forms (Hinze, 1975; Monin and Yaglom, 1975; Panofsky and Dutton, 1984).

8.7.4 Taylor's frozen turbulence hypothesis While trying to relate the spatial distribution of turbulence and related statistics to temporal statistics obtained from measurements at a ®xed point, Taylor (1938) proposed the following plausible hypothesis: If the velocity of air stream which carries the eddies is very much greater than the turbulent velocity, one may assume that the sequence of changes in u at the ®xed point are simply due to the passage of an unchanging pattern of turbulent motion over the

160

8 Fundamentals of turbulence

point, i.e., one may assume that u(t) = u(x/U) where x is measured upstream at time t = 0 from the ®xed point where u is measured.

This so-called frozen-turbulence hypothesis was originally suggested for a homogeneous ®eld of low-intensity turbulence in a uniform mean ¯ow. It implies a direct correspondence between spatial changes in a turbulence variable in the mean ¯ow direction and its temporal changes at a ®xed point, assuming an unchanging or frozen pattern of turbulence. The hypothesis is frequently used for converting temporal statistics to spatial statistics in the direction of mean ¯ow. Earlier, it provided a practical and convenient way of testing the results of the statistical theory of homogeneous and isotropic turbulence. Subsequently, the frozen-turbulence hypothesis has been extended and generalized to all turbulent variables and also to nonhomogeneous shear ¯ows. Taylor's hypothesis has been critically examined theoretically, as well as experimentally by many investigators. Two primary limitations are found on the validity of the hypothesis in the atmospheric boundary layer. The ®rst arises from the suciently large turbulence intensities, especially under unstable and convective conditions, which are found to have the apparent e€ect of increasing the e€ective advection velocity above the actual mean velocity. The validity of the hypothesis becomes particularly questionable under free convection conditions when velocity ¯uctuations are of the same order or larger than the mean velocity. The second limitation arises from the mean ¯ow shear. In the presence of shear, large eddies may not be advected at the same local mean velocity which advects small eddies, because large velocity di€erences are likely to exist across large eddies. According to a well-known theoretical criterion, only eddies with frequencies much greater than the magnitude of mean wind shear may be assumed to be transported by the local mean velocity, as implied by Taylor's hypothesis. Aside from turbulence, Taylor's hypothesis has widespread applications in atmospheric dispersion theories and models (Arya, 1999).

8.8 Applications A knowledge of the fundamentals of turbulence is essential to any qualitative or quantitative understanding of the turbulent exchange processes occurring in the PBL, as well as in the free atmosphere. The statistical description in terms of variances and covariances of turbulent ¯uctuations, as well as in terms of eddies and scales of motion, provide the bases for further quantitative analyses of observations. At the same time the basic concepts of the generation and maintenance of turbulence, of supply, transfer, and dissipation of energy, and of local similarity are the foundation stones of turbulence theory.

Problems and Exercises

161

Problems and Exercises 1. Discuss the various types of local or small-scale instability mechanisms that might be operating in the atmosphere and their possible e€ects on atmospheric motions. 2. If you have available only the upper air velocity and temperature soundings, how would you use this information to determine which layers of the atmosphere might be turbulent? 3. (a) What are the possible mechanisms of generation and maintenance of turbulence in the atmosphere? (b) What criterion may be used to determine if turbulence in a stably strati®ed layer could be maintained or if it would decay? 4. What are the special characteristics of turbulence that distinguish it from a random wave motion? 5. Compare and contrast the time and ensemble averages. Under what conditions might the two types of averages be equivalent? 6. What is the possible range of turbulence intensities in the atmospheric PBL and under what conditions might the extreme values occur? 7. Write down all the Reynolds stress components and indicate which of these have the same magnitudes. 8. The following mean ¯ow and turbulence measurements were made at a height of 22.6 m during the 1968 Kansas Field Program under di€erent stability conditions: Run no. Ri U (m s71) su (m s71) sv (m s71) sw (m s71) sy (K) uw (m2 s72) yw (m s71 K) yu (m s71 K)

19

20

43

54

18

17

23

71.89 4.89 0.83 1.02 0.65 0.48 70.081 0.185 70.064

71.15 6.20 1.04 1.07 0.73 0.61 70.111 0.273 70.081

70.54 8.06 1.27 1.26 0.73 0.56 70.244 0.188 70.175

70.13 9.66 1.02 0.91 0.64 0.25 70.214 0.072 70.129

0.08 7.49 0.73 0.56 0.48 0.14 70.118 70.029 0.068

0.12 5.50 0.45 0.36 0.28 0.15 70.043 70.016 0.039

0.22 5.02 0.20 0.17 0.10 0.18 70.005 70.005 0.023

162

8 Fundamentals of turbulence

(a) Calculate and plot turbulence intensities as functions of Ri. (b) Calculate and plot the magnitudes of correlation coecients ruw, rwy, and ruy as functions of Ri. Explain their di€erent signs under unstable and stable conditions. (c) Calculate the turbulent kinetic energy (TKE) for each run and plot the ratio 7TKE/uw as a function of Ri. (d) What conclusion can you draw about the variation of the above turbulence quantities with stability? 9. (a) Calculate Kolmogorov's microscales of length and velocity for the following values of the rate of energy dissipation observed in the atmospheric boundary layer under di€erent stability conditions and at di€erent heights in the PBL: e = 1074, 1073, 1072, and 1071 m2 s73 (b) Comment on the possible variation of Z with height in the surface layer, considering that e generally decreases with height (it is inversely proportional to z near the surface).

Chapter 9

Models and Theories of Turbulence

9.1 Mathematical Models of Turbulent Flows In Chapter 7 we introduced the equations of continuity, motion, and thermodynamic energy as mathematical expressions of the conservation of mass, momentum, and heat in an elementary volume of ¯uid. These are applicable to laminar as well as turbulent ¯ows. In the latter, all the variables and their temporal and spatial derivatives are highly irregular and rapidly varying functions of time and space. This essential property of turbulence makes all the terms in the conservation equations signi®cant, so that no further simpli®cations of them are feasible, aside from the Boussinesq approximations introduced in Chapter 7. In particular, the instantaneous equations to the Boussinesq approximations for a thermally strati®ed turbulent ¯ow, in a rotating frame of reference tied to the earth's surface, are ~ @ u~ @~ v @w ‡ ‡ ˆ0 @x @y @z @ u~ @ u~ @ u~ @ u~ ~ ‡ u~ ‡ v~ ‡ w ˆ f v~ @t @x @y @z @~ v @~ v @~ v @~ v ~ ˆ ‡ u~ ‡ v~ ‡ w @t @x @y @z

1 @ p~1 ‡ nH2 u~ r0 @x

f u~

1 @ p~1 ‡ nH2 v~ r0 @y

~ ~ ~ ~ @w @w @w @w g ~ ~ T1 ‡ u~ ‡ v~ ‡w ˆ @t @x @y @z T0

…9:1†

1 @ p~1 ~ ‡ nH2 w r0 @z

~ @ ~y @y @~ y @~ y ~ y ‡ u~ ‡ v~ ‡ w ˆ ah H 2 ~ @t @x @y @z Here, a tilde denotes an instantaneous variable which is the sum of its mean and ¯uctuating parts. 163

164

9 Models and Theories of Turbulence

No general solution to this highly nonlinear system of equations is known and there appears to be no hope of ®nding one through purely analytical methods. However, approximate numerical solutions through the use of supercomputers have been obtained for a variety of low-Reynolds-number turbulent ¯ows. For large-Reynolds-number ¯ows, such as those encountered in the atmosphere including the PBL, only the Reynolds-averaged equations obtained from a grid-volume averaging (spatial ®ltering) or an ensemble averaging of the instantaneous equations can be solved. Their numerical solutions including the various ®nite-di€erence and ®nite-element methods constitute the rapidly growing ®eld of computational ¯uid dynamics. 9.1.1 Direct numerical simulation In this age of superfast computers, it is natural to attempt direct numerical solutions of the instantaneous Navier±Stokes equations for the various turbulent ¯ows of interest using the brute force method of numerical integration on supercomputers or networks of small computers or work stations. This method has been called full-turbulence simulation or direct numerical simulation (DNS); so far it has been limited to only low-Reynolds-number turbulent ¯ows. Since the instantaneous equations describe all the details of the turbulent ¯uctuating motion, their direct numerical solution with appropriate boundary conditions should yield every detail of its complex behavior in time and space. Numerical methods are available for solving Equations (9.1) in principle, but there is a serious problem of a numerical model's ability to resolve the whole wide range of scales encountered in turbulent ¯ow. In any direct numerical simulation of turbulence, the ¯ow domain is discretized into three-dimensional grid elements. For an appropriate resolution of small-scale motions, which are mostly responsible for the dissipation of turbulence kinetic energy, the grid size must be less than the microscale Z, the characteristic length scale of such motions. For the proper resolution of large energy-containing eddies, the model domain should be larger than the largest scales (l ) of interest. Thus, the number of three-dimensional grid elements or grid points at which instantaneous equations are solved at each small time step (this is also determined by the microscale) is of the order of (l/Z)3. This number for the atmospheric boundary layer can be estimated to fall in the range of 1015±1020. The present supercomputers cannot handle more than 109 grid points for direct numerical simulation of turbulence. Thus, so far, DNS has been limited to low-Reynoldsnumber ¯ows with l/Z 5 103. Apart from the very large storage requirements, much faster computer speeds will be needed before DNS becomes practically feasible for atmospheric applications.

9.1 Mathematical Models of Turbulent Flows

165

9.1.2 Large-eddy simulation A computationally more feasible but less fundamental approach which has been used in micrometeorology and engineering ¯uid mechanics is large-eddy simulation (LES). It attempts to faithfully simulate only the scales of motion in a certain range of scales between the smallest grid size and the largest dimension of simulated ¯ow domain. The smaller subgrid scales are not resolved, but their important contributions to energy dissipation and minor contributions to turbulent transports are usually parameterized through the use of simpler subgrid scale (SGS) models (Mason, 1994). The origins of the LES technique lie in the early global weather prediction and general circulation models in which the permissible grid spacing could hardly resolve large-scale atmospheric structures. Large-eddy simulations of turbulent ¯ows, including the PBL, started with the pioneering work of Deardor€ (1970a, b, 1972a, b, 1973). The soundness of this modeling approach in simulating neutral and unstable PBLs, even in the presence of moist convection and clouds, has been amply demonstrated by subsequent studies of Deardor€ and others (Mason, 1994). A pressing need has been the extension of the LES to the nocturnal stable boundary layer, including the morning and evening transition periods, which are poorly understood. The main diculty in simulating the stably strati®ed boundary layer is that the characteristic large-eddy scale (l ) becomes small (say, 510 m) under very stable conditions that are typically encountered in the SBL over land surfaces and most of the energy transfer and other exchange processes are overly in¯uenced or dominated by subgrid scale motions. Therefore, recent LES studies of the SBL have been limited to conditions of mild or moderate stability, strong wind shears, and continuous turbulence across the SBL (Andren, 1995; Ding et al., 2000b; Kosovic and Curry, 2000). Satisfactory large-eddy simulations of the transition from the late afternoon, unstable or convective boundary layer to moderately stable boundary layer in the early evening hours have also been accomplished, especially during the early period of continuous but decaying turbulence. Similar LES studies of very stable boundary layers with frequent episodes of turbulence generation by shear and destruction by buoyancy should become possible in the near future as more powerful next-generation computers become available. A more detailed description of the large-eddy simulation approach and its applications to the PBL will be given in Chapter 13. Here, it should suce to say that LES is viewed as the most promising tool for future research in turbulence and is expected to provide a better understanding of the transport phenomena, well beyond the reach of semiempirical theories and most routine observations. Since it requires an investment of huge computer resources (there are also a few other limitations), LES will most likely remain a research tool. It would be very

166

9 Models and Theories of Turbulence

useful, of course, for generating numerical turbulence data for studying largeeddy structures, developing parameterizations for simpler models, and designing physical experiments (Wyngaard and Moeng, 1993). 9.1.3 Ensemble-averaged turbulence models Direct numerical simulation of turbulent ¯ows is based on the `primitive' Navier±Stokes equations of motion. The large-eddy simulation utilizes the same equations, but averaged over the ®nite grid volume. Numerical integrations of both systems yield highly irregular, turbulent ¯ow variables as functions of time and space. Average statistics, such as means, variances, and higher moments, are then calculated from this `simulated turbulence' data. In most applications, however, only the average statistics are needed and how they come about and other details of turbulent motions are usually of little or no concern. One may then ask, `Why not use the equations of mean motion to compute the desired average statistics?' This question was ®rst addressed by Osborne Reynolds toward the end of the nineteenth century. He also suggested some simple averaging rules or conditions and derived what came to be known as the Reynolds-averaged equations. If f and g are two independent variables or functions of variables with mean values f and g, and c is a constant, the Reynolds-averaging conditions are f ‡ g ˆ f‡ g  f cf ˆ cf; g ˆ fg Z Z   @f =@s ˆ @ f=@s; f ds ˆ fds

…9:2†

where s = x, y, z, or t. These conditions are used in deriving the equations for mean variables from those of the instantaneous variables. Since these are strictly satis®ed only by ensemble averaging, this type of averaging is generally implied in theory. The time and space averages often used in practice can satisfy the Reynolds-averaging rules only under certain idealized conditions (e.g., stationarity and homogeneity of ¯ow), which were discussed in Chapter 8. The usual procedure for deriving Reynolds-averaged equations is to substitute in Equation (9.1) u~ ˆ U ‡ u, v~ ˆ V ‡ v, etc., and take their averages using the Reynolds-averaging conditions [Equation (9.2)]. Consider ®rst the continuity equation @…U ‡ u†=@x ‡ @…V ‡ v†=@y ‡ @…W ‡ w†=@z ˆ 0

(9.3)

9.1 Mathematical Models of Turbulent Flows

167

which, after averaging, gives @U=@x ‡ @V=@y ‡ @W=@z ˆ 0

(9.4)

Subtracting Equation (9.4) from (9.3), one obtains the continuity equation for the ¯uctuating motion @u=@x ‡ @v=@y ‡ @w=@z ˆ 0

(9.5)

Thus, the form of continuity equation remains the same for instantaneous, mean, and ¯uctuating motions. This is not the case, however, for the equations of conservation of momentum and heat, because of the presence of nonlinear advection terms in those equations. Let us consider, for example, the advection terms in the instantaneous equation of the conservation of heat ~ ~y=@z† y=@x† ‡ v~…@ ~ y=@y† ‡ w…@ a~y ˆ u~…@ ~

(9.6a)

which, after utilizing the continuity equation, can also be written in the form ~~y† u~ y† ‡ …@=@y†…~ v~ y† ‡ …@=@z†…w a~y ˆ …@=@x†…~

(9.6b)

Expressing variables as sums of their mean and ¯uctuating parts in Equation (9.6b) and averaging, one obtains the Reynolds-averaged advection terms Ay ˆ

@ @ @ …UY† ‡ …VY† ‡ …WY† @x @y @z @ @ @ ‡ …uy† ‡ …vy† ‡ …wy† @x @y @z

…9:7a†

or, after using the mean continuity equation, Ay ˆ U

@Y @Y @Y @ @ @ ‡V ‡W ‡ …uy† ‡ …vy† ‡ …wy† @x @y @z @x @y @z

…9:7b†

Thus, upon averaging, the nonlinear advection terms yield not only the terms which may be interpreted as advection or transport by mean ¯ow, but also several additional terms involving covariances or turbulent ¯uxes. These latter terms are the spatial gradients (divergence) of turbulent transports.

168

9 Models and Theories of Turbulence

Following the above procedure with each component of Equation (9.1), one obtains the Reynolds-averaged equations for the conservation of mass, momentum, and heat: @U @V @W ‡ ‡ ˆ0 @x @y @z @U @U @U @U ‡U ‡V ‡W ˆ fV @t @x @y @z

1 @P1 ‡ nH2 U r0 @x @u2 @uv @uw ‡ ‡ @x @y @z

@V @V @V @V ‡U ‡V ‡W ˆ @t @x @y @z

!

1 @P1 ‡ nH2 V r0 @y ! @uv @v2 @vw ‡ ‡ @y @x @z

fU

…9:8†

@W @W @W @W g T1 ‡U ‡V ‡W ˆ @t @x @y @z T0

1 @P1 ‡ nH2 W r0 @z ! @wu @wv @w2 ‡ ‡ @x @y @z

@Y @Y @Y @Y ‡U ‡V ‡W ˆ ah H2 Y @t @x @y @z

  @uy @vy @wy ‡ ‡ @x @y @z

When these equations are compared with the corresponding instantaneous equations [Equation (9.1)], one ®nds that most of the terms (except for the turbulent transport terms) are similar and can be interpreted in the same way. However, there are several fundamental di€erences between these two sets of equations, which forces us to treat them in entirely di€erent ways. While Equation (9.1) deals with instantaneous variables varying rapidly and irregularly in time and space, Equation (9.8) deals with mean variables which are comparatively well behaved and vary only slowly and smoothly. While all the terms in the former equation set may be signi®cant and cannot be ignored a priori, the mean ¯ow equations can be greatly simpli®ed by neglecting the molecular di€usion terms outside of possible viscous sublayers and also other terms on the basis of certain boundary layer approximations and considerations of stationarity and horizontal homogeneity, whenever applicable. For example, it is easy to show that for a horizontally homogeneous and stationary PBL the equations of mean ¯ow reduce to

9.1 Mathematical Models of Turbulent Flows f …V f…U

Vg † ˆ Ug † ˆ

@uw=@z @vw=@z

169 …9:9†

An important consequence of averaging the equations of motion is the appearance of turbulent ¯ux-divergence terms which contain unknown variances and covariances. Note that in Equation (9.8) there are many more unknowns than the number of equations. While the number of extra unknowns may be reduced to one or two in certain simple ¯ow situations, the Reynoldsaveraged equations remain essentially unclosed and, hence, unsolvable. This socalled closure problem of turbulence has been a major stumbling block in developing a rigorous and general theory of turbulence. It is a consequence of the nonlinearity of the original (instantaneous) equations of motion. Many semiempirical theories and turbulence closure models have been proposed to get around the closure problem, but none of them has proved to be entirely satisfactory. The simplest gradient-transport theories, hypotheses, or relations, which are still widely used in micrometeorology, form the basis of the so-called ®rst-order closure models; these are based on the equations of mean motion. More sophisticated higher-order closure models are based on the equations of mean motion, turbulence variances, covariances, and even higher moments. Derivation of the dynamical equations for the various turbulence statistics is outside the scope of this book. It would suce to say that the fundamental closure problem remains, or even gets worse, as one includes second and higherorder moments in a turbulence closure model. The addition of the turbulence kinetic energy equation to the equations of motion forms an optimal set of dynamical equations on which most frequently used turbulence models in computational ¯uid dynamics, boundary-layer meteorology, and mesoscale meteorology are based. 9.1.4 Turbulence kinetic energy equation An approximate form of the turbulence kinetic energy equation for gradually developing atmospheric boundary layer ¯ows is   DE @U @V g @ wp ˆ uw ‡ vw wyv we ‡ e …9:10† Dt @z @z Tv0 @z r0 where E and e are mean and ¯uctuating turbulence kinetic energies per unit mass E ˆ 12 …u2 ‡ v2 ‡ w2 †

(9.11)

e ˆ 12 …u2 ‡ v2 ‡ w2 †

(9.12)

170

9 Models and Theories of Turbulence

wyv is the virtual heat ¯ux, and e is the rate of energy dissipation. The left-hand side of Equation (9.10) represents local and advective changes of E, while the various terms on the right-hand side represent shear production, buoyancy production or destruction, turbulent transport including that due to pressure ¯uctuations, and the rate of dissipation due to viscosity. The relative importance of shear generation and buoyancy production/ destruction terms for the maintenance of turbulence in the atmospheric boundary layer has already been discussed in Section 8.2. Their ratio is used to de®ne the ¯ux Richardson number gwyv Rf ˆ Tv0



 @U @V uw ‡ vw @z @z

…9:13†

which is related to the gradient Richardson number as Rf ˆ …Kh =Km †Ri

(9.14)

For the horizontally homogeneous PBL, the left-hand side of Equation (9.10) simpli®es to @E/@t, and any imbalance between the various source (e.g., shear production) and sink (e.g., dissipation and buoyancy destruction) terms will determine whether turbulence kinetic energy increases, remains constant, or decreases with time. In the unstable surface layer, both shear and buoyancy mechanisms act in concert to produce vigorous turbulence, part of which is transported upward to the mixed layer. Due to much reduced wind shears in the convective mixed layer, however, turbulence there is produced primarily by buoyancy. Although turbulence kinetic energy increases with time in response to increasing heat ¯ux and increasing PBL height in the morning hours, @E/@t is found to be an order of magnitude smaller than the individual production and dissipation terms in the TKE equation. Thus, there is an approximate balance between production, dissipation, and transport terms of that equation. In the stable boundary layer, on the other hand, the buoyancy acts in opposition to the shear production, resulting in rather weak and, sometimes, decaying turbulence. Based on their relative magnitudes, Richardson (1920) arrived at his well-known criterion of Rf 5 Rfc = 1 for the maintenance of turbulence. However, he did not include the important dissipation term in his theoretical considerations. Subsequent theoretical and experimental studies of turbulence in the SBL have yielded much lower values of Rfc = 0.2±0.5 (Arya, 1972; Stull, 1988). The criterion is better formulated in terms of the critical ¯ux Richardson number (Rfc), rather than Ric.

9.2 Gradient-transport Theories

171

9.2 Gradient-transport Theories In order to close the set of equations given by Equation (9.8) or its simpli®ed version for a given ¯ow situation, the variances and covariances must either be speci®ed in terms of other variables or additional equations must be developed. In the latter approach, the closure problem is only shifted to a higher level in the hierarchy of equations that can be developed. We will not discuss here the socalled higher-order closure schemes or models that have been developed in recent years and that have been found to have their own limitations and problems. The older and more widely used approach has been based on the assumed (hypothetical) analogy between molecular and turbulent transfers. It is called the gradient-transport approach, because turbulent transports or ¯uxes are sought to be related to the appropriate gradients of mean variables (velocity, temperature, etc.). Several di€erent hypotheses have been used in developing such relationships.

9.2.1 Eddy viscosity (di€usivity) hypothesis In analogy with Newton's law of molecular viscosity [see Equation (7.1), Chapter 7], J. Boussinesq in 1877 proposed that the turbulent shear stress in the direction of ¯ow may be expressed as t ˆ rKm …@U=@z†

(9.15)

in which Km is called the eddy exchange coecient of momentum or simply the eddy viscosity, which is analogous to the molecular kinematic viscosity n. In analogy with the more general constitutive relations [Equation (7.2)], one can also generalize Equation (9.15) to express the various Reynolds stress components in terms of mean gradients. In particular, when the mean gradients in the x and y directions can be neglected in comparison to those in the z direction (the usual boundary layer approximation), we have the simpler eddy viscosity relations for the vertical ¯uxes of momentum uw ˆ

Km …@U=@z†

vw ˆ

Km …@V=@z†

…9:16†

Similar relations have been proposed for the turbulent ¯uxes of heat, water vapor, and other transferable constituents (e.g., pollutants), which are analogous to Fourier's and Fick's laws of molecular di€usion of heat and mass.

172

9 Models and Theories of Turbulence

Those frequently used in micrometeorology are the relations for the vertical ¯uxes of heat (yw) and water vapor (qw) yw ˆ

Kh …@Y=@z†

qw ˆ

Kw …@Q=@z†

…9:17†

in which Kh and Kw are called the eddy exchange coecients or eddy di€usivities of heat and water vapor, respectively, and Q and q denote the mean and ¯uctuating parts of speci®c humidity. It should be recognized that the above gradient-transport relations are not the expressions of any sound physical laws in the same sense that their molecular counterparts are. These are not based on any rigorous theory, but only on an intuitive assumption of similarity or analogy between molecular and turbulent transfers. Under ordinary circumstances, one would expect heat to ¯ow from warmer to colder regions, roughly in proportion to the temperature gradient. Similarly, momentum and mass transfers may be expected to be proportional to and down the mean gradients. However, these expectations are not always borne out by experimental data in turbulent ¯ows, including the atmospheric boundary layer. The analogy between molecular and turbulent transfers has subsequently been found to be very weak and qualitative only. Eddy di€usivities, as determined from their de®ning relations, Equations (9.16) and (9.17), are usually several orders of magnitude larger than their molecular counterparts, indicating the dominance of turbulent mixing over molecular exchanges. More importantly, eddy di€usivities cannot simply be regarded as ¯uid properties; these are actually turbulence or ¯ow properties, which can vary widely from one ¯ow to another and from one region to another in the same ¯ow. Eddy di€usivities show no apparent dependence on molecular properties, such as mass density, temperature, etc., and have nothing in common with molecular di€usivities, except for the same dimensions. There are other limitations of the K theory which we have not mentioned so far. The basic notion of down-gradient transport implied in the theory may be questioned. There are practical situations when turbulent ¯uxes are in no way related to the local gradients. For example, in a convective mixed layer the potential temperature gradient becomes near zero or slightly positive, while the heat is transported upward in signi®cant amounts. This would imply in®nite or even negative values of Kh, indicating that K theory becomes invalid in this case. Even in other situations, the speci®cation of eddy di€usivities in a rational manner is not always easy, but quite dicult. In spite of the above limitations of the implied analogy between molecular and turbulent di€usion, Equations (9.16) and (9.17) need not be restrictive, since they replace only one set of unknowns (¯uxes) for another (eddy

9.2 Gradient-transport Theories

173

di€usivities). Some restrictions are imposed, however, when one assumes that eddy di€usivities depend on the coordinates and ¯ow parameters in some de®nite manner. This, then, constitutes a semiempirical theory that is based on a hypothesis and is subject to experimental veri®cation. The simplest assumption, which Boussinesq proposed originally, is that eddy di€usivities are constants for the whole ¯ow. It turns out that this assumption works well in free turbulent ¯ows such as jets, wakes, and mixing layers, away from any boundaries, and is often used in the free atmosphere. But when applied to boundary layers and channel ¯ows, it leads to incorrect results. In general, the assumption of constant eddy di€usivity is not applicable near a rigid surface. But here other reasonable hypotheses can be made regarding the variation of eddy di€usivity with distance from the surface. For example, a linear distribution of Km in the neutral surface layer works quite well. Suggested modi®cations of the K distribution in thermally strati®ed conditions are usually based on other theoretical considerations and empirical data, which will be discussed later. In any case, the K theory is quite useful and is widely used in micrometeorology. 9.2.2 Mixing-length hypothesis In an attempt to specify eddy viscosity as a function of geometry and ¯ow parameters, L. Prandtl in 1925 further extended the molecular analogy by ascribing a hypothetical mechanism for turbulent mixing. According to the kinetic theory of gases, momentum and other properties are transferred when molecules collide with each other. The theory leads to an expression of molecular viscosity as a product of mean molecular velocity and the mean free path length (the average distance traveled by molecules before collision). Prandtl hypothesized a similar mechanism of transfer in turbulent ¯ows by assuming that eddies or `blobs' of ¯uid (analogous to molecules) break away from the main body of the ¯uid and travel a certain distance, called the mixing length (analogous to free path length), before they mix suddenly with the new environment. If the velocity, temperature, and other properties of a blob or parcel are di€erent from those of the environment with which the parcel mixes, ¯uctuations in these properties would be expected to occur as a result of the exchanges of momentum, heat, etc. If such eddy motions occur more or less randomly in all directions, it can be easily shown that net (average) exchanges of momentum, heat, etc., will occur only in the direction of decreasing velocity, temperature, etc. In order to illustrate the above mechanism for the generation of turbulent ¯uctuations and their covariances (¯uxes), we consider the usual case of increasing mean velocity with height in the lower atmosphere (Figure 9.1).

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9 Models and Theories of Turbulence

Figure 9.1 Schematic of mean velocity in the lower atmosphere and expected correlations between the longitudinal and vertical velocity ¯uctuations due to ¯uid parcels coming from above or below mixing with the surrounding ¯uid.

The longitudinal velocity ¯uctuations at a level z may be assumed to occur as a result of mixing with the environment, at this level, of ¯uid parcels arriving from di€erent levels above and below. For example, a parcel arriving from below (level z l) will give rise to a negative ¯uctuation at level z of magnitude u ˆ U…z

l†

U…z† 

l…@U=@z†

(9.18)

associated with its positive vertical velocity (¯uctuation) w. The last approximation is based on the assumption of a linear velocity pro®le over the mixing length l, which is considered here a ¯uctuating quantity with positive values for the upward motions and negative for the downward motions of the parcel. Considering the action of many parcels arriving at z and taking the average, we obtain an expression for the momentum ¯ux uw ˆ

lw…@U=@z†

(9.19)

which is not very helpful, because there is no way of measuring l. Another mixing-length expression for uw was obtained by Prandtl, after assuming that in a turbulent ¯ow the velocity ¯uctuations in all directions are of the same order of magnitude and are related to each other, so that w

u % l…@U=@z†

(9.20)

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175

From Equations (9.18) and (9.20) one obtains uw 

l 2 …@U=@z†2

or Prandtl's original mixing-length relation uw ˆ

l 2m j@U=@zj…@U=@z†

(9.21)

in which lm is a mean mixing length, which is proportional to the root-meansquare value of the ¯uctuating length l, and the absolute value of the velocity gradient is introduced to ensure that the momentum ¯ux is down the gradient. Note that Equations (9.20) and (9.21) are derived for the unidirectional mean ¯ow in the x direction. These can be generalized to the PBL ¯ow with U and V pro®les as w  lj@V=@zj

(9.22)

uw ˆ

l 2m j@V=@zj…@U=@z†

(9.23)

vw ˆ

l 2m j@V=@zj…@V=@z†

(9.24)

Further, extensions of the mixing-length hypothesis to the vertical transfers of heat and water vapor lead to the following analogous relations for their ¯uxes: yw ˆ

lm lh j@V=@zj…@Y=@z†

(9.25)

qw ˆ

lm lw j@V=@zj…@Q=@z†

(9.26)

where lh and lw are the mixing lengths for heat and water vapor transfers, which might di€er from that of momentum. Equations (9.23)±(9.26) constitute closure relations if the various mixing lengths can be prescribed as functions of the ¯ow geometry and, possibly, other ¯ow properties. If large eddies are mainly responsible for momentum and other exchanges in a turbulent ¯ow, it can be argued that lm should be directly related to the characteristic large-eddy length scale. Some investigators make no distinction between the two, although, strictly speaking, a proportionality coecient of the order of one is usually involved. In free turbulent ¯ows and in the outer parts of boundary layer and channel ¯ows, where large-eddy size may not have signi®cant spatial variations, the assumption of constant mixing length (lm = l0) is found to be reasonable. In surface or wall layers, the large-eddy size, at least normal to the surface, varies roughly in proportion to the distance from the surface, so that mixing length is also expected to be proportional to the distance (lm * z). In the atmospheric boundary layer, the mixing-length

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9 Models and Theories of Turbulence

distribution is further expected to depend on thermal strati®cation and boundary layer thickness, and will be considered later. The mixing-length hypothesis may be used to specify eddy viscosity. From a comparison of Equations (9.16) and (9.23), it is obvious that Km ˆ l 2m j@V=@zj

(9.27)

An alternative speci®cation is obtained from a comparison of Equations (9.16) and (9.19) Km ˆ cm lm sw

(9.28)

where cm is a constant. Other parametric relations of eddy viscosity involving the product of a characteristic length scale and a turbulence velocity scale have been proposed in the literature. Similar relations are given for Kh and Kw; e.g., Kh ˆ lm lh j@V=@zj

(9.29)

Kh ˆ ch lh sw

(9.30)

or, alternatively,

Example Problem 1 In the neutral surface layer, eddy viscosity and mixing length can be expressed as Km = kzu and lm = kz. Derive an expression for wind distribution, assuming that the friction velocity u ˆ … uw†1=2 is independent of height in this constant¯ux, neutral surface layer. Solution For the neutral surface layer, the eddy viscosity relation (9.16) can be expressed as u2 ˆ Km

@U @U ˆ kzu @z @z

so that, @U u ˆ @z kz whose integration with respect to z yields Uˆ

u ln z ‡ A k

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177

where A is a constant of integration. In order to determine A we use the lower boundary condition at z = z0, U = 0 (here, one cannot use the boundary condition right at the surface, i.e., at z = 0, for various reasons). This boundary condition gives A =7(u =k) ln z0 and the wind pro®le as Uˆ

u ln…z=z0 † k

which is the well known logarithmic wind pro®le law, in which z0 is called the roughness length. The mixing-length relation (9.21) leads to the same result as u2

ˆ

l 2m

 2  2 @U 2 2 @U ˆk z @z @z

or, @U u ˆ @z kz 9.3 Dimensional Analysis and Similarity Theories Dimensional analysis and similarity considerations are extensively used in micrometeorology, as well as in other areas of science and engineering. Therefore, these analytical methods are brie¯y discussed in this section, while their applications in micrometeorology will be given in later chapters.

9.3.1 Dimensional analysis Dimensional analysis is a simple but powerful method of investigating a variety of scienti®c phenomena and establishing useful relationships between the various quantities or parameters, based on their dimensions. One can de®ne a set of fundamental dimensions, such as length [L], time [T], mass [M], etc., and express the dimensions of all the quantities involved in terms of these fundamental dimensions. A representation of the dimensions of a quantity or a parameter in terms of fundamental dimensions constitutes a dimensional formula, e.g., the dimensional formula for ¯uid viscosity is [m] = [ML71T71]. If the exponents in the dimensional formula are all zero, the parameter under consideration is dimensionless. One can form dimensionless parameters from appropriate combinations of dimensional quantities; e.g., the Reynolds number

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9 Models and Theories of Turbulence

Re = VLr/m is a dimensionless combination of ¯uid velocity V, the characteristic length scale L, density r, and viscosity m. Dimensionless groups or parameters are of special signi®cance in any dimensional analysis in which the main objective is to seek certain functional relationships between the various dimensionless parameters. There are several reasons for considering dimensionless groups instead of dimensional quantities or variables. First, mathematical expressions of fundamental physical laws are dimensionally homogeneous (i.e., all the terms in an expression or equation have the same dimensions) and can be written in dimensionless forms simply by an appropriate choice of scales for normalizing the various quantities. Second, dimensionless relations represented in mathematical or graphical form are independent of the system of units used and they facilitate comparisons between data obtained by di€erent investigators at di€erent locations and times. Third, and, perhaps, the most important reason for working with dimensionless parameters is that nondimensionalization always reduces the number of parameters that are involved in a functional relationship. This follows from the wellknown Buckingham Pi theorem, which states that if m quantities (Q1 ; Q2 ; . . . ; Qm ), involving n fundamental dimensions, form a dimensionally homogeneous equation, the relationship can always be expressed in terms of m n independent dimensionless groups (II1 ; II2 ; . . . ; IIm n ) made of the original m quantities. Thus, the dimensional functional relationship f…Q1 ; Q2 ; . . . ; Qm † ˆ 0

(9.31)

is equivalent to the dimensionless relation F…II1 ; II2 ; . . . ; IIm n † ˆ 0

(9.32)

II1 ˆ F1 …II2 ; II3 ; . . . ; IIm n †

(9.33)

or, alternatively,

In particular, when only one dimensionless group can be formed out of all the quantities, i.e., when m n ˆ 1, that group must be a constant, since it cannot be a function of any other parameters. If there are two II-groups, one must be a unique function of the other, and so on. Dimensional analysis does not give actual forms of the functions F, F1, etc., or values of any dimensionless constants that might result from the analysis. This must be obtained by other means, such as further theoretical considerations and experimental observations. It is common practice to follow dimensional analysis by a systematic experimental study of the phenomenon to be investigated.

9.3 Dimensional Analysis and Similarity Theories

179

9.3.2 Similarity theory The Buckingham Pi theorem and dimensional analysis discussed above are merely mathematical formalisms and do not deal with the physics of the problem. The actual formulation of a similarity theory involves several steps, some of which require physical intuition, other theoretical considerations, prior observational information, and possibly new experiments designed to test the theory. The ®ve steps involved in developing and testing a similarity theory are: (1) De®ne the scope of the theory with all the restrictive assumptions clearly stated. (2) Select an optimal set of relevant independent variables on which one or more variable of interest may depend. This constitutes a similarity hypothesis about the functional dependence between the various variables. Only one dependent variable is considered at a time in such a functional relationship, but one can have several functional relationships (e.g., one for each dependent variable). (3) Perform dimensional analysis after determining the number of possible independent dimensionless groups and organize the variables into dimensionless groups. (4) Express functional relationships between dimensionless groups, one of which should contain the dependent variable. These constitute the similarity relations or similarity theory predictions (one for each dependent variable). (5) Gather relevant data from previous experiments that satisfy the restrictive assumptions of the similarity theory, or perform a new experiment to test the initial similarity hypothesis and similarity theory predictions. Experimental data will tell us whether the original similarity hypothesis is correct or not. If the theory is veri®ed by experimental data, the latter can also be used to determine empirical forms of the various similarity functions by appropriate curve ®tting through data plots. The ultimate result of this ®ve-step procedure is a set of empirical equations or ®tted curves through plots of experimental data, all involving dimensionless similarity parameters. For a successful similarity theory, veri®ed by experiments, the empirical similarity relations are expected to be universal and can be used at other locations with di€erent surface characteristics and under di€erent meteorological conditions. In the ®rst step, certain restrictive assumptions are made in order to reduce the number of independent variables involved in a similarity hypothesis, so that one can reduce the number of dimensionless parameters to a minimum that is consistent with the physics. The fewer the dimensionless parameters, the more powerful are the similarity theory predictions, and the easier it is to verify them

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9 Models and Theories of Turbulence

and to determine empirical similarity relations or constants from carefully conducted experiments and observations. For example, the most commonly used simplifying assumptions in all the proposed PBL similarity theories are: (1) mean ¯ow is steady or stationary; (2) mean ¯ow is horizontally homogeneous, implying a ¯at and homogeneous surface; (3) viscosity and other molecular di€usivities are not relevant in the bulk of the PBL ¯ow outside molecular sublayers; and (4) near-surface canopy variables can be ignored in the formulation of a similarity theory for the horizontally homogeneous part of the PBL. Additional assumptions are sometimes made depending on the type of the PBL (e.g., for a barotropic PBL, geostrophic winds are independent of height) and its stability regime (e.g., under convective conditions, shear e€ects and the Coriolis parameter are ignored). In surface-layer similarity theories, the Coriolis parameter, geostrophic winds and shears, and the PBL height are all considered irrelevant. The second step, involving the selection of independent variables in the formulation of a similarity hypothesis, is the most crucial step in the development of a successful similarity theory. On the one hand, one cannot ignore any of the important variables or parameters on which the dependent variable really depends, because this might lead to a completely wrong or unphysical relationship. On the other hand, if unnecessary and irrelevant variables are included in the original similarity hypothesis, they will unnecessarily complicate the analysis and make empirical determination of the various functional relationships extremely dicult, if not impossible. In principle, experimental data should tell us if there are irrelevant variables or similarity parameters, which can be dropped from the similarity theory without any signi®cant loss of generality. It is always desirable to keep the number of independent variables to a minimum, consistent with physics. Sometimes, it may become necessary to break down the domain of the problem or phenomenon under investigation into several small subdomains, so that simpler similarity hypotheses can be formulated for each of them separately. For example, the atmospheric boundary layer is usually divided into a surface layer and an outer layer or a mixed layer for dimensional analysis and similarity considerations. The third step is straightforward, once the number of independent dimensionless groups is determined by the Buckingham's Pi theorem. But, there is always some ¯exibility and choice in formulating dimensionless parameters. For this, it is often convenient to ®rst determine the appropriate scales of length, velocity, etc., from the independent variables. Then, dimensionless parameters can be determined merely by inspection. If there are more than one length or velocity scales, their ratio forms a dimensionless similarity parameter. The similarity relations or predictions in the fourth step are simply expressions of dimensionless groups containing dependent variables as unspeci®ed functions of other dimensionless (similarity) parameters. If there is no similarity

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181

parameter that can be formed by independent variables only, the dependent parameter (II-group) must simply be a constant. Finally, a thorough experimental veri®cation of the similarity hypothesis and resulting similarity relations is necessary before a proposed similarity theory becomes widely accepted and successful. For this, experiments are conducted at di€erent locations, under more or less idealized conditions implied in the theory. However, the whole wide range of parameters should be covered. Sometimes, experimental data can verify some of the similarity relations, but not others, in which case the theory is considered only partially successful with applicability to only speci®ed variables.

9.3.3 Examples of similarity theories In order to illustrate the method and usefulness of dimensional analysis and similarity, let us consider the possible relationship for the mean potential temperature gradient (@Y=@z) as a function of the height (z) above a uniform heated surface, the surface heat ¯ux (H0), the buoyancy parameter (g/T0) which appears in the expressions for static stability and buoyant acceleration, and the relevant ¯uid properties (r and cp) in the near-surface layer when free convection dominates any mechanical mixing (this latter condition permits dropping of all shear-related parameters from consideration). To establish a functional relationship in the dimensional form f…@Y=@z; H0 ; g=T0 ; z; r; cp † ˆ 0

(9.34)

would require extensive observations of temperature as a function of height and surface heat ¯ux at di€erent times and locations (to represent di€erent types of surfaces and radiative regimes). If the relationship is to be further generalized to other ¯uids, laboratory experiments using di€erent ¯uids will also be necessary. Considerable simpli®cation can be achieved, however, if r and cp are combined with H0 into what may be called kinematic heat ¯ux H0/rcp, so that Equation (9.34) can be written as F…@Y=@z; H0 =rcp ; g=T0 ; z† ˆ 0

(9.35)

If we now use the method of dimensional analysis, realizing that only one dimensionless group can be formed from the above quantities, we obtain …@Y=@z†…H0 =rcp †

2=3

…g=T0 †1=3 z4=3 ˆ C

(9.36)

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9 Models and Theories of Turbulence

in which the left-hand side is the dimensionless group that is predicted to be a constant. The value of the constant C can be determined from only one carefully conducted experiment, although a thorough experimental veri®cation of the above relationship might require more extensive observations. The desired dimensionless group from a given number of quantities can often be formed merely by inspection. A more formal and general approach would be to write and solve a system of algebraic equations for the exponents of various quantities involved in the dimensionless group. For example, the dimensionless group formed out of all the parameters in Equation (9.35) may be assumed as II1 ˆ …@Y=@z†…H0 =rcp †a …g=T0 †b zc

(9.37)

in which we have arbitrarily assigned a value of unity to one of the indices (here, the exponent of @Y=@z), because any arbitrary power of a dimensionless quantity is also dimensionless. Writing Equation (9.37) in terms of our chosen fundamental dimensions (length, time, and temperature), we have [L0T0K0] = [KL71][KLT71]a [LT72 K71]b [L]c from which we obtain the algebraic equations 0ˆ 0ˆ

1‡a‡b‡c a 2b

0ˆ1‡a

…9:38†

b

whose solution gives a ˆ 2=3, b ˆ 1=3, and c ˆ 4=3. Substituting these values in Equation (9.37) and equating the only dimensionless group to a constant, then, yields Equation (9.36). Another approach is to ®rst formulate the characteristic scales of length, velocity, etc., from combinations of independent variables and then use these scales to normalize the dependent variables. In the case of multiple scales, their ratios form the independent dimensionless groups. In the above example of temperature distribution over a heated surface, with @Y=@z as the dependent variable and the remaining quantities as independent variables, the following scales can be formulated out of the latter:

9.3 Dimensional Analysis and Similarity Theories length: z temperature:



yf ˆ 

velocity:

uf ˆ

H0 rcp

2=3 

H0 g z rcp T0

g T0 1=3



1=3

z

1=3

183

…9:39†

Then, the appropriate dimensionless group involving the dependent variable is …z=yf †…@Y=@z†, which must be a constant, since no other independent dimensionless groups can be formed from independent variables. This procedure also leads to Equation (9.36); it is found to be more convenient to use when a host of dependent variables are functions of the same set of independent variables. For example, standard deviations of temperature and vertical velocity ¯uctuations in the free convective surface layer are given by sy =yf ˆ Cy sw =uf ˆ Cw

…9:40†

or, after substituting from Equation (9.39), sy ˆ Cy …H0 =rcp †2=3 …g=T0 †

1=3

sw ˆ Cw ‰…H0 =rcp †…g=T0 †zŠ1=3

z

1=3

…9:41†

Equations (9.41) have proved to be quite useful for the atmospheric surface layer under daytime unstable conditions and are supported by many observations (Monin and Yaglom, 1971, Chapter 5; Wyngaard, 1973) from which cy % 1:3 and Cw % 1:4. The above similarity theory was originally proposed by Obukhov and is more commonly known as the local free convection similarity theory. Local free convection similarity and scaling are not found to be applicable to horizontal velocity ¯uctuations, so that su and sv are not simply proportional to uf. Instead, they are found to be strongly in¯uenced by large-eddy motions (convective updrafts and downdrafts) which extend through the whole depth of the CBL. Since the PBL height is ignored in the local free convection similarity theory, its applicability is limited to vertical velocity and temperature ¯uctuations. If the boundary layer height (h) is also added to the list in Equation (9.35), we would have, according to the pi theorem, two independent dimensionless

184

9 Models and Theories of Turbulence

groups and the predicted functional relationship would be   @Y H0 @z rcp

2=3 

g T0

1=3

z4=3 ˆ F…z=h†

…9:42†

Similarly, sw/uf and sy =yf would be predicted to be functions of z/h. However, subsequent analysis of experimental data might indicate that the dependence of these on z/h is very weak or nonexistent and, hence, the irrelevance of h in the original hypothesis. The inclusion of h would be quite justi®ed and even necessary, however, if we were to investigate the turbulence structure of the mixed layer, or su and sv even in the surface layer. For example, the mixed-layer similarity hypothesis proposed by Deardor€ (1970b) states that turbulence structure in the convective mixed layer depends on z, g/T0, H0/rcp, and h. Then, the relevant mixed-layer similarity scales are: length: h temperature: velocity:



   H0 2=3 gh T ˆ rcp T0  1=3 H0 g W ˆ h rcp T0

1=3

…9:43†

The corresponding similarity predictions are that the dimensionless structure parameters su =W , sw =W , etc., must be some unique function of z/h. This mixed-layer similarity theory has proved to be very useful in describing turbulence and di€usion in the convective boundary layer (Arya, 1999). As the number of independent variables and parameters is increased in a similarity hypothesis, not only the number of independent dimensionless groups increases, but also the possible combinations of variables in forming such groups become large. The possibility of experimentally determining their functional relationships becomes increasingly remote as the number of dimensionless groups increases beyond two or three. A generalized PBL similarity theory will have to include all the possible factors in¯uencing the PBL under the whole range of conditions encountered and, hence, might be too unwieldy for practical use. Simpler PBL similarity theories and scaling will be described in Chapter 13, while the surface layer similarity theory/scaling will be discussed in more detail in Chapters 10±12. Example Problem 2 In a convective boundary layer during the midday period when T0 = 300 K, H0 = 500 W m72, and h = 1500 m, calculate and compare the standard

Problems and Exercises

185

deviations of vertical velocity and temperature ¯uctuations at the 10 m and 100 m height levels. Solution Using the local free-convection similarity relations (9.41) with Cy ˆ 1:3, Cw ˆ 1:4, and rcp ˆ 1200 J K71 m73, the computed values of sw and sy at the two heights in the convective surface layer (z 5 0.1h = 150 m) are as follows: Height, z (m)

sw (m s71)

sy (K)

10 100

0.72 1.55

1.05 0.49

Note that sw increases and sy decreases by a factor of 101/3 % 2.15 as the height increases from 10 to 100 m.

9.4 Applications The theories and models of turbulence discussed in this chapter are widely used in micrometeorology. Some of the speci®c applications are as follows: . Calculating the mean structure (e.g., vertical pro®les of mean velocity and temperature) of the PBL. . Calculating the turbulence structure (e.g., pro®les of ¯uxes, variances of ¯uctuations, and scales of turbulence) of the PBL. . Providing plausible theoretical explanations for turbulent exchange and mixing processes in the PBL. . Providing suitable frameworks for analyzing and comparing micrometeorological data from di€erent sites. . Suggesting simple methods of estimating turbulent ¯uxes from mean pro®le observations.

Problems and Exercises 1. Compare and contrast the instantaneous and the Reynolds-averaged equations of motion for the PBL.

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9 Models and Theories of Turbulence

2. (a) What boundary layer approximations are often used for simplifying the Reynolds-averaged equations of motion for the PBL? (b) What are the other simplifying assumptions commonly used in micrometeorology and the conditions in which they may not be valid? 3. Show, step by step, how Equation (9.8) is reduced to Equation (9.9) for a horizontally homogeneous and stationary PBL. 4. (a) Write down the mean thermodynamic energy equation for a horizontally homogeneous PBL and discuss the rationale for retaining the time-tendency term in the same. (b) Describe a method for estimating the diurnal variation of surface heat ¯ux from hourly soundings of temperature in the PBL. 5. Show that with the assumption of a constant eddy viscosity the equations of mean motion in the PBL become similar to those for the laminar Ekman layer and, hence, discuss some of the limitations of the above assumption. 6. If the surface layer turbulence under neutral stability conditions is characterized by a large-eddy length scale l, which is proportional to height, and a velocity scale u , which is independent of height, suggest the plausible expressions of mixing length and eddy viscosity in this layer. 7. A similarity hypothesis proposed by von Karman states that mixing length in a turbulent shear ¯ow depends only on the ®rst and second spatial derivatives of mean velocity (@U=@z, @ 2 U=@z2 ) in the direction of shear. Suggest an expression for mixing length which is consistent with the above similarity hypothesis. 8. (a) In a neutral, barotropic PBL, ageostrophic velocity components (U Ug and V Vg ) may be assumed to depend only on the height z above the surface, the friction velocity scale u , and the Coriolis parameter f. On the basis of dimensional analysis suggest the appropriate similarity relations for velocity distribution and the PBL height h. (b) If the PBL height in Problem 8(a) above was determined by a low-level inversion, with its base at zi, what should be the corresponding form of similarity relations?

Problems and Exercises

187

9. (a) In a convective boundary layer during the midday period when T0 = 310 K, H0 = 300 W m72, and h = 1000 m, calculate and compare the standard deviations of velocity and temperature ¯uctuations at the heights of 10, 50, and 100 m. (b) Show that, in the convective surface layer, the local free convection similarity relations can also be expressed in terms of mixed-layer similarity scales as sy =T ˆ Cy …z=h†

1=3

; sw =W ˆ Cw …z=h†1=3

Chapter 10

Near-neutral Boundary Layers

10.1 Velocity-pro®le Laws Strictly neutral stability conditions are rarely encountered in the atmosphere. However, during overcast skies and strong surface geostrophic winds, the atmospheric boundary layer may be considered near-neutral, and simpler theoretical and semiempirical approaches developed for neutral boundary layers by ¯uid dynamists and engineers can be used in micrometeorology. One must recognize, however, that, unlike in unidirectional ¯at-plate boundary layer and channel ¯ows, the wind direction in the PBL changes with height in response to the Coriolis force due to the earth's rotation. Therefore, the wind distribution in the PBL is expressed in terms of either wind speed and direction, or the two horizontal components of velocity (see Figures 6.5±6.8). 10.1.1 The power-law pro®le Measured velocity distributions in ¯at-plate boundary layer and channel ¯ows can be represented approximately by a power-law expression U=Uh ˆ …z=h†m

(10.1)

which was originally suggested by L. Prandtl with an exponent m = 1/7 for smooth surfaces. Here, h is the boundary layer thickness or half-channel depth. Since, wind speed does not increase monotonically with height up to the top of the PBL, a slightly modi®ed version of Equation (10.1) is used in micrometeorology: U=Ur ˆ …z=zr †m

(10.2)

where Ur is the wind speed at a reference height zr, which is smaller than or equal to the height of wind speed maximum; a standard reference height of 10 m is commonly used. 188

10.1 Velocity-pro®le Laws

189

The power-law pro®le does not have a sound theoretical basis, but frequently it provides a reasonable ®t to the observed velocity pro®les in the lower part of the PBL, as shown in Figure 10.1. The exponent m is found to depend on both the surface roughness and stability. Under near-neutral conditions, values of m range from 0.10 for smooth water, snow and ice surfaces to about 0.40 for welldeveloped urban areas. Figure 10.2 shows the dependence of m on the roughness length or parameter z0, which will be de®ned later. The exponent m also increases with increasing stability and approaches one (corresponding to a linear pro®le) under very stable conditions. The value of the exponent may also depend, to some extent, on the height range over which the power law is ®tted to the observed pro®le.

Figure 10.1 Comparison of observed wind speed pro®les at di€erent sites (z0 is a measure of the surface roughness) under di€erent stability conditions with the power-law pro®le. [Data from Izumi and Caughey (1976).]

190

10 Near-neutral Boundary Layers

Figure 10.2 Variations of the power-law exponent with the roughness length for nearneutral conditions. [Data from Counihan (1975). Reprinted with permission from Atmospheric Environment. Copyright # (1975), Pergamon Journals Ltd.]

Note that the power-law wind pro®le would be appropriate only for the lower part of the PBL in which wind speed increases monotonically with height. At higher levels, wind speed may be speci®ed di€erently (e.g., constant or linearly varying with height), depending on stability and other conditions (Arya, 1999, Chapter 4). The power-law velocity pro®le implies a power-law eddy viscosity (Km) distribution in the lower part of the boundary layer, in which the momentum ¯ux may be assumed to remain nearly constant with height, i.e., in the constant stress layer. It is easy to show that Km/Kmr = (z/zr)n

(10.3)

with the exponent n ˆ 1 m, is consistent with Equation (10.2) in the surface layer. Equations (10.2) and (10.3) are called conjugate power laws and have been used extensively in theoretical formulations of atmospheric di€usion, including transfers of heat and water vapor from extensive uniform surfaces (Sutton, 1953). In such formulations eddy di€usivities of heat and mass are assumed to be equal or proportional to eddy viscosity, and thermodynamic energy and di€usion equations are solved for prescribed velocity and eddy di€usivity pro®les in the above manner. When Equations (10.2) and (10.3) are used above the constant stress layer, however, the conjugate relationship n ˆ 1 m need not be satis®ed; this relationship may be too restrictive even in the surface layer.

10.1 Velocity-pro®le Laws

191

10.1.2 The logarithmic pro®le law Let us focus our attention now to the neutral surface layer over a ¯at and uniform surface in which Coriolis force can be ignored and the momentum ¯ux may be considered constant, independent of height. Furthermore, we also exclude from our consideration any viscous sublayer that may exist over a smooth surface, or the canopy or roughness sublayer in which the ¯ow is very likely to be disturbed by individual roughness elements. For the remaining fully turbulent, horizontally homogeneous surface layer, a simple similarity hypothesis can be used to obtain the velocity distribution, namely, the mean wind shear @U=@z is dependent only on the height z above the surface (more appropriately, above a suitable reference plane near the surface), the surface drag, and the ¯uid density, i.e., @U=@z ˆ f …z; t0 ; r† ˆ f …z; t0 =r†

(10.4)

An implied assumption in this similarity hypothesis is that the in¯uence of other possible parameters, such as the surface roughness, horizontal pressure gradients (geostrophic winds), and the PBL height, is fully accounted for in t0 , which then determines the velocity gradients in the surface layer. Note that it is appropriate to combine t0 and r into their ratio t0/r, which represents the kinematic momentum ¯ux, because the dependent variable does not contain any mass dimension. The only characteristic velocity scale given by the above surface layer similarity hypothesis is the so-called friction velocity u  …t0 =r†1=2 , and the only characteristic length scale is z. Then, from dimensional analysis it follows that the dimensionless wind shear …z=u †…@U=@z† ˆ const: ˆ 1=k

(10.5)

where k is called von Karman's constant. The above similarity relation has been veri®ed by many observed velocity pro®les in laboratory boundary layer, channel, and pipe ¯ows, as well as in the near-neutral atmospheric surface layer. The von Karman constant is presumably a universal constant for all surface or wall layers. However, it is an empirical constant with a value of about 0.40; it has not been possible to determine it with an accuracy better than 5% (Hogstrom, 1985). Note that Equation (10.5) also follows from the mixing length and eddy viscosity hypotheses, if one assumes that l = kz and/or Km = kzu in the constant-¯ux surface layer. Integration of Equation (10.5) with respect to z gives the well-known logarithmic velocity pro®le law, U=u ˆ …1=k† ln…z=z0 †

(10.6)

192

10 Near-neutral Boundary Layers

in which z0 has been introduced as a dimensional constant of integration and is commonly referred to as the roughness parameter or roughness length. The physical signi®cance of z0 and its relationship to surface roughness characteristics will be discussed later. For a comparison of the power-law and logarithmic wind pro®les, Equation (10.6) can be written as U=Ur ˆ 1 ‡ ln…z=zr †= ln…zr =z0 †

(10.7)

Figure 10.3 compares the plots of U/Ur versus z/zr, according to Equations (10.2) and (10.7), for di€erent values of m and zr/z0, respectively. There is no exact correspondence between these two pro®le parameters, because the two pro®le shapes are di€erent. An approximate relationship can be obtained by matching the velocity gradients, in addition to the mean velocities, at the reference height zr:   d…ln U† z @U zr ˆ ˆ mˆ d…ln z† U @z zˆzr z0

1

…10:8†

Figure 10.3 Comparison of hypothetical wind pro®les following the power-law Equation (10.2) (Ð) and the log-law Equation (10.7) (± ± ±) with their parameters related by Equation (10.8).

10.1 Velocity-pro®le Laws

193

Even with this type of matching, the two pro®les deviate more and more as z deviates farther and farther from the matching height zr. The theoretical relationship given by Equation (10.8) between the power-law exponent m and the surface roughness parameter is also compared with observations in Figure 10.2. For di€erent data sets represented in that ®gure, zr lies in the range between 10 and 100 m. The logarithmic pro®le law has a sounder theoretical and physical basis than the power-law pro®le, especially within the neutral surface layer. The former implies linear variations of eddy viscosity and mixing length with height, irrespective of the surface roughness, while the power-law pro®le implies the incorrect and unphysical, nonlinear behavior of Km and lm, even close to the surface. Many observed wind pro®les near the surface under near-neutral conditions have con®rmed the validity of Equation (10.6) up to heights of 20±200 m, depending on the PBL height. Figure 10.4 illustrates some of the observed wind

Figure 10.4 Comparison of the observed wind pro®les in the neutral surface layer of Wangara Day 43 with the log law [Equation (10.6)] (solid lines). [Data from Clarke et al. (1971).]

194

10 Near-neutral Boundary Layers

pro®les during the Wangara Experiment which was conducted over a low grass surface in southern Australia. These plots of ln z versus U also illustrate how the best-®tted straight line through observed data points can be used to estimate the surface-layer parameters u and z0. Note that, according to Equation (10.6), the slope of the line must be equal to k=u , while its intercept on the ln z axis where U = 0 must be equal to ln z0. Thus estimates of slope and intercept of the best®tted line yield u and z0. If there are several wind pro®les available at the same site, one can determine z0 as the geometric mean of the values obtained from individual pro®les. In the engineering ¯uid mechanics literature, Equation (10.6) is referred to as the law of the wall, which has been found to hold in all kinds of pipe and channel ¯ows, as well as in boundary layers. There, the distinction is made between aerodynamically smooth and rough surfaces. A surface is considered aerodynamically smooth if the small-scale surface protuberances or irregularities are suciently small to allow the formation of a laminar or viscous sublayer in which surface protuberances are completely submerged. If small-scale surface irregularities are large enough to prevent the formation of a viscous sublayer, the surface is considered aerodynamically rough. It is found from laboratory measurements that the thickness of the laminar sublayer in which the velocity pro®le is linear is about 5n=u . For atmospheric ¯ows the sublayer thickness would be of the order of 1 mm or less, while surface protuberances are generally larger than 1 mm. Therefore, almost all natural surfaces are aerodynamically rough; the only exceptions might be smooth ice, snow, and mud ¯ats, as well as water surfaces under weak winds (U10 m 5 3 m s71). The law of the wall (velocity-pro®le law) for an aerodynamically smooth surface is usually expressed in the form (Monin and Yaglom, 1971, Chapter 3) U 1 u z 1  u z  ˆ ln ‡ 5:1 % ln u k n k 0:13n

…10:9†

in which the characteristic height scale of the viscous sublayer (n=u ) is used in place of z0 for a rough surface, and the numerical constant is evaluated from laboratory measurements in smooth ¯at-plate boundary layer and channel ¯ows. A comparison of Equations (10.6) and (10.9) shows that, if the former is used to represent wind pro®les over a smooth surface, one should expect the roughness parameter to decrease with increasing u , according to the relation z0 % 0.13n=u . For a rigid, aerodynamically rough surface, however, z0 is expected to remain constant, independent of u . Some surfaces are neither smooth nor completely rough, but fall into a transitional roughness regime. In nature, lake, sea and ocean surfaces may fall into this transition between

10.1 Velocity-pro®le Laws

195

smooth and rough surfaces at moderate wind speeds (2.5 5 U10 5 7.5 m s71) (Kitaigorodski, 1970). The transfer of momentum from the atmosphere to the ocean or sea surface is of considerable importance to meteorologists, as well as to physical oceanographers. A part of this momentum goes into the generation of surface waves, while the remaining portion is responsible for setting up drift currents and turbulence in the upper layers of the ocean. The relative partitioning of momentum between these di€erent modes of motion in water depends on such factors as the stage of wave growth, duration and fetch of winds, etc., and is not easy to determine. But the total momentum exchange can be determined from measurements in the atmospheric surface layer, well above the level of highest wave crests. When one considers extending the `law of the wall' similarity to air ¯ow over a moving water surface, the e€ects of both the surface drift currents and surface waves should be taken into account. The surface current, which is typically 3±5% of the wind speed at 10 m, but can be much stronger in certain regions (e.g., the Gulf Stream, Kuroshio, and other ocean currents), can easily be taken into account by considering the air motion relative to that of surface water (U = Ua 7 Uw). The in¯uence of moving surface waves on mean wind pro®le in the surface layer is much more dicult to discern.

10.1.3 The velocity-defect law Above the surface layer, the Coriolis e€ects become signi®cant and it is more appropriate to consider the deviations of actual velocity from the geostrophic velocity. For ¯at-plate boundary layer and channel ¯ows, similarity considerations for the deviations of velocity from the ambient velocity lead to the socalled velocity-defect law (Monin and Yaglom, 1971, Chapter 3) …U

U1 †=u ˆ F…z=h†

(10.10)

The analogous geostrophic departure laws proposed for the neutral barotropic PBL are (Tennekes, 1982): …U

Ug †=u ˆ Fu … fz=u †

…V

Vg †=u ˆ Fv … fz=u †

…10:11†

These are based on the similarity hypothesis for the barotropic, neutral PBL that departures in mean velocity components from their geostrophic values are functions of z, u and f only. This similarity hypothesis was originally proposed

196

10 Near-neutral Boundary Layers

by Kazanski and Monin (1961). It also implies that the neutral PBL height must be proportional to u =f, i.e., h ˆ cu =j f j where c is an empirical constant with estimated values between 0.2 and 0.3. A satisfactory con®rmation of Equation (10.11) and empirical determination of the similarity functions Fu and Fv has been made from atmospheric observations under near-neutral stability conditions. A strictly neutral, stationary, and barotropic PBL, which is not constrained by any inversion from above, is so rare in the atmosphere that relevant observations of the same do not exist. The averages of wind pro®les measured under slightly unstable and slightly stable conditions appear to follow the above similarity scaling (Clarke, 1970; Nicholls, 1985). Some laboratory simulations of the neutral Ekman layer in a rotating wind tunnel ¯ow have also con®rmed the same. Of course, there is plenty of experimental support for the validity of the velocity-defect law [Equation (10.10)] in nonrotating ¯ows. Example Problem 1 The following mean wind speeds were measured over a short grass surface under near-neutral stability conditions: z (m): U (m s 1 ):

0:5 7:82

1 8:66

2 9:54

4 10:33

8 11:22

16 12:01

(a) Determine the roughness length and friction velocity from a plot of ln z versus U. (b) Using appropriate similarity relations, estimate eddy viscosity and mixing length at 10 and 100 m. Solution (a) Plotting ln z versus U and drawing the best-®tted regression line through the data points, or simply regressing ln z against U in a programmable calculator, one obtains ln z = 0.824U 7 7.15 Comparing this with Equation (10.6), which can be rewritten as ln z ˆ

k U u

ln z0

we obtain k=u = slope of the line = 0.824 s m71, or u = 0.40/0.824 = 0.485 m s71,

10.2 Surface Roughness Parameters

197

and ln z0 =77.15, so that z0 % 7.9 6 1074 m = 0.079 mm. (b) In the neutral surface layer, eddy viscosity and mixing length are given by Km = kzu ; lm = kz from which the following values can be estimated at the desired height levels (10 and 100 m): z (m)

Km (m2 s71)

lm (m)

10 100

1.94 19.40

4.0 40

10.2 Surface Roughness Parameters The aerodynamic roughness of a ¯at and uniform surface may be characterized by the average height (h0) of the various roughness elements, their areal density, characteristic shapes, and dynamic response characteristics (e.g., ¯exibility and mobility). All the above characteristics would be important if one were interested in the complex ¯ow ®eld within the roughness or canopy layer. There is not much hope for a generalized and, at the same time, simple theoretical description of such a three-dimensional ¯ow ®eld in which turbulence dominates over the mean motion. In theoretical and experimental investigations of the fully developed surface layer, however, the surface roughness is characterized by only one or two roughness characteristics which can be empirically determined from wind-pro®le observations. 10.2.1 Roughness length The roughness length parameter z0 introduced in Equation (10.6) is one such characteristic. In practice, z0 is determined from the least-square ®tting of Equation (10.6) through the wind-pro®le data, or by graphically plotting ln z versus U and extrapolating the best-®tted straight line down to the level where U = 0; its intercept on the ordinate axis must be ln z0. One should note, however, that this is only a mathematical or graphical procedure for estimating z0 and that Equation (10.6) is not expected to describe the actual wind pro®le

198

10 Near-neutral Boundary Layers

below the tops of roughness elements. An assumption implied in the derivation 5 z. of (10.6) is that z0 5 Empirical estimates of the roughness parameter for various natural surfaces can be ordered according to the type of terrain (Figure 10.5) or the average height of the roughness elements (Figure 10.6). Although z0 varies over ®ve orders of magnitude (from 1075 m for very smooth water surfaces to several meters for forests and urban areas), the ratio z0/h0 falls within a much narrower range (0.03±0.25) and increases gradually with increasing height of roughness elements. For uniform sand surfaces Prandtl and others have suggested a value of z0/h0 % 1/30. An average value of z0/h0 = 0.15 has been determined for various crops and grasslands; the same may also be used for many other natural surfaces (Plate, 1971). Figure 10.5 gives some typical values of z0 for di€erent types of surfaces and terrains. Figure 10.6 relates z0 to h0 for di€erent types of crop canopies. Wind-pro®le measurements from ®xed towers in shallow coastal waters and from stable ¯oating buoys (ships are not suitable for this, as they present too much of an obstruction to the air ¯ow) over open oceans also follow the logarithmic law [Equation (10.6)] under neutral stability conditions. However, the roughness parameter (z0) determined from this is found to be related to both the wind and wave ®elds in a rather complex manner and can vary over a very wide range (say, 1075±1072 m). There have been a number of semiempirical and theoretical attempts at relating z0 to the friction velocity, as well as to the average height, the phase speed, and the stage of development of surface waves. The simplest and still the most widely used relationship is that proposed by Charnock (1955) on the basis of dimensional arguments z0 ˆ a…u2 =g†

(10.12)

where a is an empirical constant. The basic assumptions implied in Charnock's hypothesis (z0 is uniquely determined by u and g) are that the winds are blowing steadily and long enough for the wave ®eld to be in complete equilibrium with the wind ®eld, independent of fetch, and that the surface is aerodynamically rough. Individual estimates of z0 plotted as a function of u2 =g invariably show large scatter, because they usually pertain to di€erent fetches and di€erent stages of wave development and also because of large errors in the experimental determination of z0. However, when the available data sets are block averaged, a de®nite trend of z0 increasing in proportion to u2 =g does emerge. Figure 10.7 shows such block-averaged data compiled from some 33 experimental sets of wind pro®le and drag data (Wu, 1980). Note that Charnock's relation is veri®ed, particularly for large values of z0 and u2 =g, and a % 0.018. Empirical

10.2 Surface Roughness Parameters

199

Figure 10.5 Typical values, or range of values, of the surface roughness parameter for di€erent types of terrain. [From tables by the Royal Aeronautical Society (1972).]

200

10 Near-neutral Boundary Layers

Figure 10.6 Relationship between the aerodynamic roughness parameter and the average vegetation height. [From Plate (1971).]

estimates of a by other investigators have ranged between 0.01 and 0.035 (Garratt, 1992). The simple state of a€airs assumed in Charnock's hypothesis rarely exists over lakes and oceans. More frequently, the wave ®eld is not in equilibrium with local winds, but depends on the strength, fetch, and time history of the wind ®eld. At low wind speeds, the surface can also become aerodynamically smooth, or be in a transitional regime between smooth and rough. A comprehensive treatment of sea-surface roughness at di€erent development stages of windgenerated waves is given by Kitaigorodski (1970). The fundamental parameter indicating the stage of wave development, in his theory, is the ratio C0/u , where C0 is the phase speed of the most dominant (corresponding to the peak in the wave-height spectrum) waves. It is shown that only in the equilibrium stage of wave development is z0 proportional to u2 =g. Furthermore, on the basis of analogy with the aerodynamic roughness of a rigid surface, z0 is assumed proportional to h0 for a completely rough surface; proportional to n=u for an aerodynamically smooth surface; and a function of both h0 and n=u in the transitionally smooth or rough regime. Kitaigorodski has suggested plausible expressions for the roughness parameter for the various combinations of roughness and wave regimes, some of them involving still unknown empirical

10.2 Surface Roughness Parameters

201

Figure 10.7 Observed roughness parameter of sea surface as a function of u2 =g compared with Charnock's (1955) formula. [After Wu (1980).]

constants or functions. For the fully-developed wave ®eld, a simple interpolation relation z0 ˆ a

u2 n ‡ 0:13 u g

…10:13†

has been suggested (Smith et al., 1996). At low wind speeds (say U10 5 7 m s71) and in the early stages of wave development, the sea surface appears to be remarkably smooth, before waves start to break. In fact, the aerodynamic roughness of even a fully developed sea under a tropical storm is observed to be less than that of mown grass. This is largely due to the fact that waves are generated by and move along with the wind and do not o€er as much resistance to the ¯ow as would an immobile surface with similar shape. There is also strong evidence suggesting that seasurface drag is primarily due to small ripples and wavelets with phase speeds less than the near-surface wind speed, and that large waves make only a minor contribution to the drag.

202

10 Near-neutral Boundary Layers

Di€erent surface roughness regimes mentioned earlier are found to occur over oceans. In the absence of swell, the sea surface is found to be aerodynamically smooth during calm and weak winds (U10 5 2.5 m s71), transitionally rough in moderate winds (2.5  U10  7.5 m s71), and completely rough in strong winds (U10 4 7.5 m s71). The above criteria should be considered only approximate and limited to fully developed (equilibrium) wind-generated waves. The data points corresponding to small values of u2 =g in Figure 10.7 probably belong to the transitional roughness regime. 10.2.2 Displacement height For very rough and undulating surfaces, the soil±air or water±air interface may not be the most appropriate reference datum for measuring heights in the surface layer. The air ¯ow above the tops of roughness elements is dynamically in¯uenced by the ground surface as well as by individual roughness elements. Therefore, it may be argued that the appropriate reference datum should lie somewhere between the actual ground level and the tops of roughness elements. In practice, the reference datum is determined empirically from wind-pro®le measurements in the surface layer under near-neutral stability conditions. The modi®ed logarithmic wind-pro®le law used for this purpose is U=u ˆ …1=k† ln‰…z0

d0 †=z0 Š

(10.14)

in which d0 is called the zero-plane displacement or displacement height and z0 is the height measured above the ground level. For a plane surface, d0 is expected to lie between zero and h0, depending on the areal density of roughness elements. The displacement height may be expected to increase with increasing roughness density and approach a value close to h0 for very dense canopies in which the ¯ow within the canopy might become stagnant, or independent of the air ¯ow above the canopy. These expectations are indeed borne out by empirical estimates of d0 for vegetative canopies (see Figure 10.8) which indicate that the appropriate datum for windpro®le measurements over most vegetative canopies is displaced above the ground level by 70±80% of the average height of vegetation. The zero-plane displacement in an urban boundary layer can similarly be expected to be a large fraction of the average building height. After the mean height of roughness elements, the next important characteristic of roughness morphology is the ratio l ˆ Af =Ah , where Af is the total frontal area of roughness elements and Ah is the horizontal area covered by them. As the frontal area density (l) increases, the ratio d0/h0 may be expected to increase monotonically to its maximum value of one. The ratio z0/h0 is also

10.3 Surface Drag and Resistance Laws

203

Figure 10.8 Relationship between the zero-plane displacement and average vegetation height for di€erent types of vegetation. [After Stanhill (1969).]

expected to increase and attain its maximum value at some optimum value of l 5 1. Wind tunnel experiments with regular arrays of simulated roughness elements have con®rmed these expectations (McDonald et al., 1998). Methods of determining surface roughness parameters from an analysis of surface morphometry of urban areas have also been suggested (Grimmond and Oke, 1999). Figure 10.9 shows a veri®cation of Equation (10.14) against observed wind pro®les over bare and vegetative surfaces, using appropriate estimates of z0 and d0 for each surface. 10.3 Surface Drag and Resistance Laws A direct measurement of the wind drag or stress on a relatively smooth surface can be made with an appropriate drag plate of a nominal diameter of 1±2 m and surface representative of the terrain around it (Bradley, 1968). The larger the surface roughness, the more dicult it becomes for the drag plate to represent (simulate) such a roughness. Therefore, this method of drag measurement is limited to ¯at and uniform surfaces which are bare or have low vegetation. In most other cases, and also when drag-plate measurements cannot be made, the surface stress is determined indirectly from wind measurements in the surface layer.

204

10 Near-neutral Boundary Layers

Figure 10.9 Comparison of observed velocity pro®les over crops with the log law [Equation (10.14)]. [From Plate (1971).]

10.3.1 Surface drag law If mean winds are observed at a standard reference height zr, the surface stress can be determined using a drag relation t0 ˆ rCD U 2r

(10.15)

in which CD is a dimensionless drag coecient which depends on the surface roughness (more appropriately, on the ratio zr/z0) and atmospheric stability near the surface. In particular, for near-neutral stability, the drag coecient is given by CDN ˆ k2 =‰ln…zr =z0 †Š2

(10.16)

which follows from Equations (10.6) and (10.15). Note that for a given zr, CDN increases with increasing surface roughness. For a standard reference height of 10 m, values of CDN range from 1.0 6 1073 for large lake and ocean surfaces at moderate wind speeds (U10 5 6 m s71) to 7.5 6 1073 for tall crops and moderately rough (z0 5 0.1 m) surfaces. For very rough surfaces (e.g., forests and urban areas), however, a reference height of 10 m would not be adequate; it must be at least 1.5 times the height of roughness elements.

10.3 Surface Drag and Resistance Laws

205

There has been considerable interest in knowing the dependence of CDN over ocean surfaces on wind speed, sea state, and other factors. To this end, many experimental determinations of CDN have been made by di€erent investigators, using di€erent measurement techniques. Early determinations were largely based on rather crude estimates of the tilt of water surface and the geostrophic departure of ¯ow in the PBL, and are not considered very reliable. Later, the sea-surface drag was determined by using the more reliable eddy correlation, gradient, and pro®le methods. A comprehensive review of the existing data on CD up to 1975 was made by Garratt (1977). After applying the appropriate corrections for atmospheric stability and actual observation height, the neutral drag coecient CDN, referred to the standard height of 10 m, was obtained as a function of wind speed at 10 m (U10). Figure 10.10(a) shows the block-averaged values of CDN + 1 standard deviation (indicated by vertical bars) for intervals of U10 of 1 m s71, based on the eddy correlation method (closed circles) and wind-pro®le method (open circles). The number of data points used for averaging in each 1 m s71 interval are also indicated above the abscissa axis (the top line refers to closed circles and the bottom line to open circles). Note that there are too few data points to give reliable estimates of CDN at high wind speeds (U10 4 15 m s71). However, some investigators have inferred the surface drag in hurricanes, using the geostrophic departure method. Others have measured CDN at high wind speeds in laboratory wind ¯ume experiments. Figure 10.10(b) shows that there is a remarkable similarity between the hurricane data and wind ¯ume data (usually referred to a height of 10 cm, assuming a scaling factor of 100 :1), despite the large expected errors in the former and dissimilarity of wave ®elds in the two cases. Considering the overall block-averaged data for the whole wide range of wind speeds represented in Figure 10.10, there is no doubt that CDN increases with wind speed. The trend appears to be quite consistent with Charnock's formula [Equation (10.12)], which, in conjunction with Equations (10.15) and (10.16) yields 1=2

ln CDN ‡ kC DN ˆ ln…gz=aU 2r †

(10.17)

After determining the best-®tting value of a % 0:0144 with k = 0.41 (for the more commonly used value of k = 0.4, one would obtain a % 0:017, which is closer to the value indicated in Figure 10.7), from a regression of 1=2 ln CDN ‡ kC DN on ln U10, Equation (10.17) is represented in Figures 10.10a and b. In the completely rough regime (U10 4 7 m s71), in particular, Charnock's relation is in good agreement with the experimental data. The variation of CDN with U10 can also be approximated by a simple linear relation (Garratt, 1977) CDN ˆ …0:75 ‡ 0:067U10 †  10

3

(10.18)

206

10 Near-neutral Boundary Layers

Figure 10.10 Neutral drag coecient as a function of wind speed at 10 m height compared with Charnock's formula [Equation (10.17), indicated by arrows in (a) and (b)] with a = 0.0144. Block-averaged values are shown for (a) 1 m s71 intervals, based on eddy correlation and pro®le methods, and (b) 5 m s71 intervals, based on geostrophic departure method and wind ¯ume simulation experiments. [After Garratt (1977).]

10.3 Surface Drag and Resistance Laws

207

in which U10 is to be expressed in meters per second. A linear relationship, similar to Equation (10.18), is also suggested by other measurements of drag in gale-force winds (Smith, 1980). There is some controversy over the variation of CDN with U10 at low and moderate wind speeds (say, U10 5 7.5 m s71). At very low wind speeds (U10 5 2.5 m s71), observations from laboratory ¯ume experiments and from over lakes and oceans indicate a slight tendency for CDN to decrease with the increase in wind speed, as would be expected for an aerodynamic smooth surface. But sea surface is rarely completely smooth; ripples and wavelets generated by weak to moderate winds give it a moderate roughness, which is transitional between smooth and completely rough regimes. Some investigators have argued that the dependence of CDN on U10 should be weak, if any, in this transitional regime (2.5 5 U10 5 7.5 m s71). The data of Figure 10.10 do suggest a weaker dependence of CDN on U10 in this range. Other oceanic data have even indicated a nearly constant value of CDN % 1.2 6 1073 in the moderate range of wind speeds (Kraus, 1972). Few investigators have sought to investigate the dependence of CDN on wind direction, fetch, and sea-state parameters, such as C0 =u (Kitaigorodski, 1970; SethuRaman, 1978; Smith, 1980). There is only a slight tendency for the drag coecient to decrease with increasing fetch. It has also been argued that the coecient a in Charnock's formula should decrease with increasing C0 =u and may reach a constant asymptotic value only at the equilibrium stage of wave development. This would imply that CDN should also decrease with increasing C0 =u . An observational evidence in support of such a trend is given by SethuRaman (1978), who also shows dramatic changes in CDN in response to a rapid shift in wind direction.

10.3.2 Geostrophic drag relations Alternative parameterizations of the surface drag use the surface geostrophic wind speed (G0), or the actual wind speed at the top of the PBL, instead of the near-surface wind in expressing the surface stress, e.g., t0 ˆ rCD G20

(10.19)

in which the value of CD is expected to be smaller than that in Equation (10.15). 1=2 The ratio u =G0 ˆ C D is often called the geostrophic drag coecient and denoted by cg. For the neutral PBL, cg can be expressed as a function of the

208

10 Near-neutral Boundary Layers

surface Rossby number, G0/fz0, using the so-called geostrophic resistance laws (Tennekes, 1982): 1 Ug =u ˆ ‰ln…u =fz0 † k Vg =u ˆ B=k

AŠ

…10:20†

where A and B are similarity constants. An elegant derivation of the above resistance laws from the asymptotic matching of the neutral surface layer and the PBL similarity wind-pro®le relations (10.6) and (10.11) is given by Tennekes (1982). The similarity constants have been empirically determined from the observed wind pro®le and geostrophic winds under near-neutral conditions. Earlier estimates from observations over land surfaces showed fairly large scatter, probably due to the large sensitivity of A and B to the stability and baroclinicity of the PBL (Sorbjan, 1989). More recently, Nicholls (1985) used the mean ¯ow and turbulence data obtained over the ocean during the Joint Air±Sea Interaction Experiment (JASIN) to estimate A % 1.4 and B % 4.2. From Equations (10.19) and (10.20), the geostrophic drag coecient can be expressed as cg ˆ k‰…ln cg ‡ ln Ro

A†2 ‡ B2 Š

1=2

(10.21)

where Ro = G0/fz0 is the surface Rossby number. Thus, cg is a decreasing function of Ro; its evaluation and plot is left as an exercise for the reader.

10.4 Turbulence Turbulence in a neutrally strati®ed boundary layer is entirely of mechanical origin and depends on the surface friction and vertical distribution of wind shear. From similarity considerations discussed in Section 10.1, the primary velocity scale is u and normalized standard deviations of velocity ¯uctuations (su =u , sv =u , and sw =u ) must be constants in the surface layer and some functions of the normalized height fz=u in the outer layer. Surface layer observations from di€erent sites indicate that under near-neutral conditions su/u % 2.4, sv/u % 1.9, and sw =u % 1.3 (Panofsky and Dutton, 1984, Chapter 7). Observed values of turbulence intensity (su =U) are shown in Figure 10.11 as a function of the roughness parameter. These are compared with the theoretical relation su/U = l/ln(z/z0), which is based on su =u = 2.5 and the logarithmic wind-pro®le law. The above theoretical relation seems to overestimate turbulence intensities over very rough surfaces. Counihan (1975) has suggested an

10.4 Turbulence

209

Figure 10.11 Variation of turbulence intensity in the near-neutral surface layer with the roughness length. [After Counihan (1975). Reprinted with permission from Atmospheric Environment. Copyright # (1975), Pergamon Journals Ltd.]

alternative empirical relationship between su =U and log z0, which provides a better ®t to the observed data in Figure 10.11. Measurements of turbulence in the upper part of the near-neutral PBL are limited to few experiments in the marine PBL (Pennell and LeMone, 1974; Nicholls, 1985) but numerical model results indicate approximately exponential decrease of su =u , etc., with height. For rough estimates of turbulent ¯uctuations, we suggest the following relations: su =u ˆ 2:4 exp j

afz=u j

sv =u ˆ 1:9 exp j

afz=u j

sw =u ˆ 1:3 exp j

afz=u j

…10:22†

in which the constant a may depend on the empirical constant c in the PBL height relationship h ˆ cu =j f j. If we de®ne the PBL height as the level where turbulent kinetic energy or momentum ¯ux reduces to 10% of its value at the surface, then, we ®nd that a = 1.15/c (e.g., for c = 0.25, a = 4.6). Note that the above relations (10.22) imply the following expression for the TKE: E ˆ 5:5u2 exp j

2afz=u j

(10.23)

Example Problem 2 For the wind pro®le data given in the Example Problem 1 for a site in southern Australia (latitude = 34.58S), estimate the following turbulence parameters at the indicated heights in the neutral PBL:

210

10 Near-neutral Boundary Layers

(a) (b) (c) (d)

the drag coecient for zr = 10 m; the vertical turbulence intensities at 10 and 100 m; turbulence kinetic energy (TKE) near the surface; TKE at 100, 200, and 500 m.

Solution (a) We can estimate the wind speed at 10 m U10 ˆ

u z 0:485 ln ˆ …ln 10 ‡ 7:15† ˆ 11:46 m s 0:40 k z0

1

Then, CD = (u /U10)2 = 1.79 6 1073. (b) In the surface layer, sw = 1.3u = 0.63 m s71 The vertical turbulence intensity is de®ned as iw = sw/U and depends on the wind speed at a given height. At z = 10 m, iw ˆ

sw ˆ 0:055 U10

iw ˆ

sw ˆ 0:044 U100

Similarly, at z = 100 m,

(c) Turbulence kinetic energy near the surface is given by E ˆ 12…u2 ‡ v2 ‡ w2 †

ˆ 12‰…2:4†2 ‡ …1:9†2 ‡ …1:3†2 Šu2

% 5:5u2 ˆ 1:29 m2 s

2

(d) For estimating TKE at higher levels in the PBL, we use Equation (10.23) with a = 4.6. For the given latitude f =734.58 f ˆ 2O sin f ˆ

0:826  10

4

s

1

Problems and Exercises

211

Then, the estimated values of TKE at various heights are: z (m): E (m2 s 2 ):

100 1:11

200 0:95

500 0:59

The estimated PBL height h ˆ 0:25u =j f j % 1468 m.

10.5 Applications Semiempirical wind-pro®le laws and other relations discussed in this chapter may have the following practical applications: . . . . .

Determining the wind energy potential of a site. Estimating wind loads on tall buildings and other structures. Calculating dispersion of pollutants in very windy conditions. Characterizing aerodynamic surface roughness. Parameterizing the surface drag in large-scale atmospheric models, as well as in wave height and storm surge formulations.

Problems and Exercises 1. Show that in the constant-stress surface layer, the power-law wind pro®le [Equation (10.2)] implies a power-law eddy viscosity pro®le [Equation (10.3)] and that the two exponents are related as n ˆ 1 m. 2. Derive the logarithmic wind-pro®le law on the basis of von Karman's mixing-length hypothesis, which implies lm ˆ k

@U @z



@2U @z2

in the constant-stress surface layer. 3. Using the logarithmic wind-pro®le law, plot the expected wind pro®les (using a linear height scale) to a height of 100 m for the following combinations of the roughness parameter (z0) and the mean wind speed at 100 m: (a) z0 = 0.01 m; U100 = 5, 10, 15, and 20 m s71; (b) U100 = 10 m s71; z0 = 1073, 1072, 1071, and 1 m. Also calculate for each pro®le the friction velocity u and the drag coecient CD for a reference height of 10 m, and show their values on the graphs.

212

10 Near-neutral Boundary Layers

4. The following mean velocity pro®les were measured during the Wangara Experiment conducted over a uniform short-grass surface under near-neutral stability conditions: Height (m): U (m s 1 ):

0:5 1 4:91 5:44

2 4 6:06 6:64

8 7:17

16 7:71

(a) Determine the roughness parameter and the surface stress for the above observation period. Take r = 1.25 kg m73. (b) Estimate eddy viscosity, mixing length, and turbulence intensities at 5 and 50 m heights. 5. If the sea surface behaves like a smooth surface at low wind speeds (U10 < 2:5 m s71), like a completely rough surface at high wind speeds (U10 4 7.5 m s71), and like a transitionally rough surface at speeds in between, express and plot the drag coecient CD as a function of U10 in the range 0.5  U10 m s71  50, assuming the following: (a) z0 = 0.13 n/u in the aerodynamically smooth regime; (b) z0 = 0.02 u2 /g in the aerodynamically rough regime; (c) z0 = 2 6 1074 m in the transitional regime. 6. (a) Derive the geostrophic drag relations (10.20) for the neutral PBL from the matching of the geostrophic departure laws (10.11) with the logarithmic wind pro®le law. Note that both the mean velocity and its vertical gradient should be matched in the surface layer. (b) Using Equation (10.21) with A = 1.4 and B = 4.2 for the neutral barotropic PBL, calculate and plot the geostrophic drag coecient (cg) as a function of the surface Rossby number (Ro) for the wide range of expected values of the latter (say, Ro = 104±109). 7. (a) Plot su =u , sv =u , sw =u , and E=u2 as functions of the normalized height fz=u in the neutral PBL, using a = 4.6. (b) After comparing the computed pro®les of E=u2 versus fz=u for a = 3.5 and 5.5, discuss the sensitivity of the TKE pro®le to the empirical constant c in the PBL height relation h ˆ cju =f j. In order to remove the dependence on a or c, express E=u2 as an exponential function of z/h.

Chapter 11

Thermally Strati®ed Surface Layer

11.1 The Monin±Obukhov Similarity Theory In the previous chapter, we showed that the atmospheric surface layer under neutral conditions is characterized by a logarithmic wind pro®le and nearly uniform (with respect to height) pro®les of momentum ¯ux and standard deviations of turbulent velocity ¯uctuations. It was also mentioned that the neutral stability condition is an exception rather than the rule in the lower atmosphere. More often, the turbulent exchange of heat between the surface and the atmosphere leads to thermal strati®cation of the surface layer and, to some extent, of the whole PBL, as shown in Chapter 5. It has been of considerable interest to micrometeorologists to ®nd a suitable theoretical or semiempirical framework for a quantitative description of the mean and turbulence structure of the strati®ed surface layer. The Monin±Obukhov similarity theory has provided the most suitable and acceptable framework for organizing and presenting micrometeorological data, as well as for extrapolating and predicting certain micrometeorological information where direct measurements are not available. 11.1.1 The similarity hypothesis The basic similarity hypothesis ®rst proposed by Monin and Obukhov (1954) is that in a horizontally homogeneous surface layer the mean ¯ow and turbulence characteristics depend only on the four independent variables: the height above the surface z, the surface drag t0/r, the surface kinematic heat ¯ux H0/rcp, and the buoyancy variable g/T0 (this appears in the expressions for buoyant acceleration and static stability given earlier). This is an extension of the earlier hypothesis for the neutral surface layer. The simplifying assumptions implied in this similarity hypothesis are that the ¯ow is horizontally homogeneous and quasistationary, the turbulent ¯uxes of momentum and heat are constant (independent of height), the molecular exchanges are insigni®cant in comparison with turbulent exchanges, the rotational e€ects can be ignored in the 213

214

11 Thermally Strati®ed Surface Layer

surface layer, and the in¯uence of surface roughness, boundary layer height, and geostrophic winds is fully accounted for through t0/r. Because the four independent variables in the M±O similarity hypothesis involve three fundamental dimensions (length, time, and temperature), according to Buckingham's theorem, one can formulate only one independent dimensionless combination out of them. The combination traditionally chosen in the Monin±Obukhov similarity theory is the buoyancy parameter z = z/L where Lˆ

u3 =‰k…g=T0 †…H0 =rcp †Š

(11.1)

is an important buoyancy length scale, known as the Obukhov length after its originator. In his 1946 article in an obscure Russian journal (for the English translation, see Obukhov, 1946/1971), Obukhov introduced L as `the characteristic height (scale) of the sublayer of dynamic turbulence'; he also generalized the semiempirical theory of turbulence to the strati®ed atmospheric surface layer and used this approach to describe theoretically the mean wind and temperature pro®les in the surface layer in terms of the fundamental stability parameter z/L. One may wonder about the physical signi®cance of the Obukhov length (L) and the possible range of its values. From the de®nition, it is clear that the values of L may range from 1 to 1, the extreme values corresponding to the limits of the heat ¯ux approaching zero from the positive (unstable) and the negative (stable) side, respectively. A more practical range of |L|, corresponding to fairly wide ranges of values of u and |H0| encountered in the atmosphere, is shown in Figure 11.1. Here, we have assumed a typical value of kg/T0 = 0.013 m s72 K71, which may not di€er from the actual value of this by more than 15%. In magnitude |L| represents the thickness of the layer of dynamic in¯uence near the surface in which shear or friction e€ects are always important. The wind shear e€ects usually dominate and the buoyancy e€ects essentially remain insigni®cant in the lowest layer close to the surface (z 5 5 |L|). On the other hand, buoyancy e€ects may dominate over shear-generated turbulence for z4 4 |L|. Thus, the ratio z/L is an important parameter measuring the relative importance of buoyancy versus shear e€ects in the strati®ed surface layer, similar to the Richardson number (Ri) introduced earlier. The negative sign in the de®nition of L is introduced so that the ratio z/L has the same sign as Ri. Later on, it will be shown that Ri and z/L are intimately related to each other, even though they have di€erent distributions with respect to height in the surface layer (z/L obviously varies linearly with height, showing the increasing importance of buoyancy with height above the surface).

11.1 The Monin±Obukhov Similarity Theory

215

Figure 11.1 Obukhov's buoyancy length as a function of the friction velocity and the surface heat ¯ux.

11.1.2 The M±O similarity relations The following characteristic scales of length, velocity, and temperature are used to form dimensionless groups in the Monin±Obukhov similarity theory: Length scales: Velocity scale: Temperature scale:

z and L u y =7H0/rcpu

216

11 Thermally Strati®ed Surface Layer

The similarity prediction that follows from the M±O hypothesis is that any mean ¯ow or average turbulence quantity in the surface layer, when normalized by an appropriate combination of the above-mentioned scales, must be a unique function of z/L only. Thus, a number of similarity relations can be written for the various quantities (dependent variables) of interest. For example, with the x axis oriented parallel to the surface stress or wind (the appropriate surface layer coordinate system), the dimensionless wind shear and potential temperature gradient are usually expressed as …kz=u †…@U=@z† ˆ fm …z† …kz=y †…@Y=@z† ˆ fh …z†

…11:2†

in which the von Karman constant k is introduced only for the sake of convenience, so that fm(0) = 1, and fm(z) and fh(z) are the basic universal similarity functions which relate the constant ¯uxes t ˆ t0 ˆ ru2 H ˆ H0 ˆ rcp u y

…11:3†

to the mean gradients in the surface layer. It is easy to show from the de®nition of Richardson number and Equations (11.1)±(11.2) that Ri ˆ zfh …z†=f2m …z†

(11.4)

which relates Ri to the basic stability parameter z = z/L of the M±O similarity theory. The inverse of Equation (11.4), namely, z = f(Ri) is often used to determine z and, hence, the Obukhov length L from easily measured gradients of velocity and temperature at one or more heights in the surface layer. Then, Equations (11.2) and (11.3) can be used to determine the ¯uxes of momentum and heat, knowing the empirical forms of the similarity functions fm(z) and fh(z) from carefully conducted experiments. Other properties of ¯ow which relate turbulent ¯uxes to the local mean gradients, such as the exchange coecients of momentum and heat and the corresponding mixing lengths, can also be expressed in terms of fm(z) and fh(z). For example, it is easy to show that Km =…kzu † ˆ fm1 …z† Kh =…kzu † ˆ fh 1 …z† Kh =Km ˆ fm …z†=fh …z†

…11:5†

11.2 Empirical Forms of Similarity Functions

217

which can be used to prescribe the eddy di€usivities or their ratio Kh/Km in the strati®ed surface layer. 11.2 Empirical Forms of Similarity Functions As mentioned earlier, a similarity theory based on a particular similarity hypothesis and dimensional analysis can only suggest plausible functional relationships between certain dimensionless parameters. It does not tell anything about the forms of those functions, which must be determined empirically from accurate observations during experiments speci®cally designed for this purpose. Experimental veri®cation is also required, of course, for the original similarity hypothesis or its consequences (e.g., predicted similarity relations). Following the proposed similarity theory by Monin and Obukhov (1954), a number of micrometeorological experiments have been conducted at di€erent locations (ideally, ¯at and homogeneous terrain with uniform and low roughness elements) and under fair-weather conditions (to satisfy the conditions of quasistationarity and horizontal homogeneity) for the expressed purpose of verifying the M±O similarity theory and for accurately determining the forms of the various similarity functions. Early experiments lacked direct and accurate measurements of turbulent ¯uxes, which could be estimated only after making some a priori assumptions about ¯ux-pro®le relations. The 1953 Great Plains Experiment at O'Neill, Nebraska, was one of the earliest concerted e€orts in observing the PBL in which micrometeorological measurements were made by several di€erent groups, using di€erent instrumentation and techniques (Lettau and Davidson, 1957). But, here too, as in subsequent Australian and Russian experiments in the 1960s, the ¯ux measurements su€ered from severe instrument-response problems and were not very reliable. Still, these experiments could verify the general validity of M±O similarity theory and give the approximate forms of the similarity functions fm(z) and fh(z). Empirical evaluations of the M±O similarity functions based on several micrometeorological experiments conducted in southern Australia have been reported by Dyer (1967), Dyer and Hicks (1970), Webb (1970), and Hicks (1976). Garratt and Hicks (1990) presented an overview of these experiments and their results. Perhaps the best micrometeorological experiment conducted so far, for the speci®c purpose of determining the M±O similarity functions, was the 1968 Kansas Field Program (Izumi, 1971). A 32 m tower located in the center of a 1 mile2 ®eld of wheat stubble (h0 & 0.18 m) was instrumented with fast-response cup anemometers, thermistors, resistance thermometers, and three-dimensional sonic anemometers at various levels. These were used to determine mean velocity and temperature gradients, as well as the momentum and heat ¯uxes using the eddy correlation method. In addition, two large drag plates installed

218

11 Thermally Strati®ed Surface Layer

nearby were used to measure the surface stress directly. Both heat and momentum ¯uxes were found to be constant with height, within the limits of experimental accuracy (+20%, for ¯uxes). The ¯ux-pro®le relations derived from the Kansas Experiment are discussed in detail by Businger et al. (1971). The generally accepted forms of fm(z) and fh(z) on the basis of the various micrometeorological experiments are ( fm ˆ ( fh ˆ

…1 g1 z† 1 ‡ b1 z; a…1 g2 z† a ‡ b2 z;

1=4

1=2

for z < 0 (unstable) for z  0 (stable)

…11:6†

for z < 0 (unstable) for z  0 (stable)

…11:7†

There remain some di€erences, however, in the estimated values of the constants, a, b1, b2, g1, and g2 in the above expressions as obtained by di€erent investigators. The main causes of these di€erences are the unavoidable measurement errors and deviations from the ideal conditions assumed in the theory (Yaglom, 1977; Hogstrom, 1988). The originally estimated values from the Kansas Experiment were (Businger et al., 1971): a = 0.74;

b1 = b2 = 4.7;

g1 = 15;

g2 = 9

(11.8)

The corresponding formulas [Equations (11.6) and (11.7) for fm and fh with the above empirical values of constants are shown in Figures 11.2 and 11.3 together with the observed Kansas data. These data are characterized by relatively small scatter, compared with other micrometeorological data sets used for evaluating the M±O similarity functions (Hogstrom, 1988). Still, they were probably subjected to signi®cant errors due to distortion of air ¯ow around instrument boxes mounted on the tower. In particular, unusually low values of k = 0.35 and a = 0.74, estimated from the Kansas data, have been attributed to likely ¯ow distortion errors. The low value of the von Karman constant, compared to its traditional value of 0.40, generated some controversial hypotheses or explanations on the possible dependence of k on roughness Reynolds number. But more de®nitive experimental determinations of the same in the 1980s did not ®nd any such dependence and have con®rmed its traditional value (Hogstrom, 1988). More accurate estimates of a from both the ®eld and laboratory experiments have also indicated that a may not be signi®cantly di€erent from one (Hogstrom, 1988). Still, the precise forms of the M±O similarity functions fm(z) and fh(z) remain far from being settled (Hogstrom, 1996).

11.2 Empirical Forms of Similarity Functions

219

Figure 11.2 Dimensionless wind shear as a function of the M±O stability parameters. [Kansas data from Izumi (1971).]

In view of the above-mentioned uncertainties in measurements and in the determination of empirical similarity functions or constants, the following simpler ¯ux-pro®le relations can be recommended for most practical applications in which great precision is not warranted: fh ˆ f2m ˆ …1 15z† fh ˆ fm ˆ 1 ‡ 5z;

1=2

;

for z < 0 for z  0

…11:9†

220

11 Thermally Strati®ed Surface Layer

Figure 11.3 Dimensionless potential temperature gradient as a function of the M±O stability parameters. [Kansas data from Izumi (1971).]

These deviate from the Kansas relations [Equations (11.6)±(11.8)] only slightly, but have the advantage of relating z to Ri more simply and explicitly as z ˆ Ri; Ri ; zˆ 1 5Ri

for Ri < 0 for 0  Ri  0:2

These are veri®ed against the Kansas data in Figure 11.4.

…11:10†

11.2 Empirical Forms of Similarity Functions

221

Figure 11.4 The Richardson number as a function of the M±O stability parameters. Equation (11.10) (Ð) is compared with Kansas data. [After Businger et al. (1971).]

After substituting from Equations (11.9) and (11.10) into Equation (11.5), the eddy di€usivities of heat and momentum can also be expressed as functions of the Richardson number Km ˆ kzu Kh ˆ kzu

( (

…1 1

15Ri†1=4 ; 5Ri;

…1 15Ri† 1 5Ri;

for Ri < 0 for 0  Ri  0:2

1=2

for Ri < 0 for 0  Ri  0:2

…11:11†

These are represented in Figure 11.5. Note that the eddy di€usivities of heat and momentum are equal under stably strati®ed conditions; they decrease rapidly with increasing stability and vanish as the Richardson number approaches its critical value of Ric = 1/b % 0.2. There is a question about the validity of the above-mentioned empirically estimated similarity functions under extremely unstable (approaching free convection) and extremely stable (approaching critical Ri) conditions.

222

11 Thermally Strati®ed Surface Layer

Figure 11.5 Variation of the dimensionless eddy di€usivities of heat and momentum with Richardson number according to Equations (11.11).

Micrometeorological data used in the determination of the above M±O similarity functions have generally been limited to the moderate stability range of 75 5 z 5 2, and, as such, are strictly valid in this range. As a matter of fact, in conditions approaching free convection (7z 4 4 1, or z ? 7?), the above expressions for fh and Kh are not quite consistent with the local free convection similarity relations for @Y/@z and Kh derived in Chapter 9. In order for them to 4 1. Some of the proposed be consistent one must have fh * (7z)71/3 for 7z 4 forms of fh satisfy this condition, but do not represent the data well, especially for 7z 5 2. On the other side of strong stability conditions, there is some evidence of a linear velocity pro®le in the lower part of the PBL, including the surface layer, which appears to be consistent with the M±O relation for fm(z) for z 4 4 1. There is also other experimental evidence suggesting the limiting value of z % 1 for the validity of the M±O relation fm = 1 + bz (Webb, 1970; Hicks, 1976). In any case, strong stability conditions usually occur at nighttime under clear skies and weak winds. Because the temperature pro®le under such conditions is strongly in¯uenced by longwave atmospheric radiation, which is ignored in the M±O similarity theory, the temperature ®eld may not follow the M±O similarity. Experimental estimates of both fm(z) and fh(z) indicate that these functions become increasingly ¯atter as z increases above 0.5 (Hicks, 1976; Holtslag, 1984).

11.3 Wind and Temperature Pro®les

223

11.3 Wind and Temperature Pro®les Integration of Equations (11.2) with respect to height yields the velocity and potential temperature pro®les in the form U=u ˆ …1=k†‰ln…z=z0 † cm …z=L†Š …Y Y0 †=y ˆ …1=k†‰ln…z=z0 † ch …z=L†Š

…11:12†

in which Y0 is the extrapolated temperature at z = z0 and cm and ch are di€erent similarity functions related to fm and fh, respectively, as cm ch

z L z L

Z ˆ

z0 =L

Z ˆ

z=L

z=L

z0 =L

dz z

‰1

fm …z†Š

‰1

dz fh …z†Š z

…11:13†

For smooth and moderately rough surfaces, z0/L is usually quite small and the integrands in Equation (11.13) are well behaved at small z, so that the lower limit of integration can essentially be replaced by zero. With this approximation, the c functions can be determined for any appropriate forms of f functions. For example, corresponding to Equation (11.9), we obtain z z ; for  0 L L "   #  1 ‡ x2 1‡x 2 cm ˆ ln 2 2

cm ˆ ch ˆ

5

  1 ‡ x2 ; ch ˆ 2 ln 2

for

2 tan

1

p x‡ ; 2

for

z